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Hexagon - Wikipedia

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1Regular hexagon

2Parameters

3Point in plane

4Symmetry

Toggle Symmetry subsection

4.1A2 and G2 groups

5Dissection

6Related polygons and tilings

Toggle Related polygons and tilings subsection

6.1Self-crossing hexagons

7Hexagonal structures

8Tesselations by hexagons

9Hexagon inscribed in a conic section

Toggle Hexagon inscribed in a conic section subsection

9.1Cyclic hexagon

10Hexagon tangential to a conic section

11Equilateral triangles on the sides of an arbitrary hexagon

12Skew hexagon

Toggle Skew hexagon subsection

12.1Petrie polygons

13Convex equilateral hexagon

Toggle Convex equilateral hexagon subsection

13.1Polyhedra with hexagons

14Gallery of natural and artificial hexagons

15See also

16References

17External links

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Hexagon

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From Wikipedia, the free encyclopedia

Shape with six sides

For the crystal system, see Hexagonal crystal family.

For other uses, see Hexagon (disambiguation).

"Hexagonal" redirects here. For the FIFA World Cup qualifying tournament in North America, see Hexagonal (CONCACAF).

Regular hexagonA regular hexagonTypeRegular polygonEdges and vertices6Schläfli symbol{6}, t{3}Coxeter–Dynkin diagramsSymmetry groupDihedral (D6), order 2×6Internal angle (degrees)120°PropertiesConvex, cyclic, equilateral, isogonal, isotoxalDual polygonSelf

In geometry, a hexagon (from Greek ἕξ, hex, meaning "six", and γωνία, gonía, meaning "corner, angle") is a six-sided polygon.[1] The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.

Regular hexagon[edit]

A regular hexagon has Schläfli symbol {6}[2] and can also be constructed as a truncated equilateral triangle, t{3}, which alternates two types of edges.

A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle).

The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals

{\displaystyle {\tfrac {2}{\sqrt {3}}}}

times the apothem (radius of the inscribed circle). All internal angles are 120 degrees. A regular hexagon has six rotational symmetries (rotational symmetry of order six) and six reflection symmetries (six lines of symmetry), making up the dihedral group D6. The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that a triangle with a vertex at the center of the regular hexagon and sharing one side with the hexagon is equilateral, and that the regular hexagon can be partitioned into six equilateral triangles.

Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane (three hexagons meeting at every vertex), and so are useful for constructing tessellations. The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a regular triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral.

A step-by-step animation of the construction of a regular hexagon using compass and straightedge, given by Euclid's Elements, Book IV, Proposition 15: this is possible as 6

{\displaystyle =}

2 × 3, a product of a power of two and distinct Fermat primes.When the side length AB is given, drawing a circular arc from point A and point B gives the intersection M, the center of the circumscribed circle. Transfer the line segment AB four times on the circumscribed circle and connect the corner points.

Parameters[edit]

R = Circumradius; r = Inradius; t = side length

The maximal diameter (which corresponds to the long diagonal of the hexagon), D, is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the inscribed circle (separation of parallel sides, flat-to-flat distance, short diagonal or height when resting on a flat base), d, is twice the minimal radius or inradius, r. The maxima and minima are related by the same factor:

cos

⁡

(

∘

)

{\displaystyle {\frac {1}{2}}d=r=\cos(30^{\circ })R={\frac {\sqrt {3}}{2}}R={\frac {\sqrt {3}}{2}}t}

and, similarly,

{\displaystyle d={\frac {\sqrt {3}}{2}}D.}

The area of a regular hexagon

≈

2.598

≈

3.464

≈

0.6495

≈

0.866

{\displaystyle {\begin{aligned}A&={\frac {3{\sqrt {3}}}{2}}R^{2}=3Rr=2{\sqrt {3}}r^{2}\\[3pt]&={\frac {3{\sqrt {3}}}{8}}D^{2}={\frac {3}{4}}Dd={\frac {\sqrt {3}}{2}}d^{2}\\[3pt]&\approx 2.598R^{2}\approx 3.464r^{2}\\&\approx 0.6495D^{2}\approx 0.866d^{2}.\end{aligned}}}

For any regular polygon, the area can also be expressed in terms of the apothem a and the perimeter p. For the regular hexagon these are given by a = r, and p

{\displaystyle {}=6R=4r{\sqrt {3}}}

, so

⋅

≈

3.464

{\displaystyle {\begin{aligned}A&={\frac {ap}{2}}\\&={\frac {r\cdot 4r{\sqrt {3}}}{2}}=2r^{2}{\sqrt {3}}\\&\approx 3.464r^{2}.\end{aligned}}}

The regular hexagon fills the fraction

≈

0.8270

{\displaystyle {\tfrac {3{\sqrt {3}}}{2\pi }}\approx 0.8270}

of its circumscribed circle.

If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumcircle between B and C, then PE + PF = PA + PB + PC + PD.

It follows from the ratio of circumradius to inradius that the height-to-width ratio of a regular hexagon is 1:1.1547005; that is, a hexagon with a long diagonal of 1.0000000 will have a distance of 0.8660254 between parallel sides.

Point in plane[edit]

For an arbitrary point in the plane of a regular hexagon with circumradius

{\displaystyle R}

, whose distances to the centroid of the regular hexagon and its six vertices are

{\displaystyle L}

and

{\displaystyle d_{i}}

respectively, we have[3]

(

)

{\displaystyle d_{1}^{2}+d_{4}^{2}=d_{2}^{2}+d_{5}^{2}=d_{3}^{2}+d_{6}^{2}=2\left(R^{2}+L^{2}\right),}

(

)

{\displaystyle d_{1}^{2}+d_{3}^{2}+d_{5}^{2}=d_{2}^{2}+d_{4}^{2}+d_{6}^{2}=3\left(R^{2}+L^{2}\right),}

(

)

{\displaystyle d_{1}^{4}+d_{3}^{4}+d_{5}^{4}=d_{2}^{4}+d_{4}^{4}+d_{6}^{4}=3\left(\left(R^{2}+L^{2}\right)^{2}+2R^{2}L^{2}\right).}

{\displaystyle d_{i}}

are the distances from the vertices of a regular hexagon to any point on its circumcircle, then [3]

(

∑

)

∑

{\displaystyle \left(\sum _{i=1}^{6}d_{i}^{2}\right)^{2}=4\sum _{i=1}^{6}d_{i}^{4}.}

Symmetry[edit]

Example hexagons by symmetry

r12regular

d6isotoxal

g6directed

p6isogonal

g2generalparallelogon

The six lines of reflection of a regular hexagon, with Dih6 or r12 symmetry, order 12.

The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Full symmetry of the regular form is r12 and no symmetry is labeled a1.

The regular hexagon has D6 symmetry. There are 16 subgroups. There are 8 up to isomorphism: itself (D6), 2 dihedral: (D3, D2), 4 cyclic: (Z6, Z3, Z2, Z1) and the trivial (e)

These symmetries express nine distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order.[4] r12 is full symmetry, and a1 is no symmetry. p6, an isogonal hexagon constructed by three mirrors can alternate long and short edges, and d6, an isotoxal hexagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction. It can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. g2 hexagons, with opposite sides parallel are also called hexagonal parallelogons.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g6 subgroup has no degrees of freedom but can be seen as directed edges.

Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation. Other hexagon shapes can tile the plane with different orientations.

p6m (*632)

cmm (2*22)

p2 (2222)

p31m (3*3)

pmg (22*)

pg (××)

r12

Dih6

Dih2

Dih1

A2 and G2 groups[edit]

A2 group roots

G2 group roots

The 6 roots of the simple Lie group A2, represented by a Dynkin diagram , are in a regular hexagonal pattern. The two simple roots have a 120° angle between them.

The 12 roots of the Exceptional Lie group G2, represented by a Dynkin diagram are also in a hexagonal pattern. The two simple roots of two lengths have a 150° angle between them.

Dissection[edit]

6-cube projection

12 rhomb dissection

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into 1⁄2m(m − 1) parallelograms.[5] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. This decomposition of a regular hexagon is based on a Petrie polygon projection of a cube, with 3 of 6 square faces. Other parallelogons and projective directions of the cube are dissected within rectangular cuboids.

Dissection of hexagons into three rhombs and parallelograms

Rhombs

Parallelograms

Regular {6}

Hexagonal parallelogons

Square faces

Rectangular faces

Cube

Rectangular cuboid

Related polygons and tilings[edit]

A regular hexagon has Schläfli symbol {6}. A regular hexagon is a part of the regular hexagonal tiling, {6,3}, with three hexagonal faces around each vertex.

A regular hexagon can also be created as a truncated equilateral triangle, with Schläfli symbol t{3}. Seen with two types (colors) of edges, this form only has D3 symmetry.

A truncated hexagon, t{6}, is a dodecagon, {12}, alternating two types (colors) of edges. An alternated hexagon, h{6}, is an equilateral triangle, {3}. A regular hexagon can be stellated with equilateral triangles on its edges, creating a hexagram. A regular hexagon can be dissected into six equilateral triangles by adding a center point. This pattern repeats within the regular triangular tiling.

A regular hexagon can be extended into a regular dodecagon by adding alternating squares and equilateral triangles around it. This pattern repeats within the rhombitrihexagonal tiling.

Regular{6}

Truncatedt{3} = {6}

Hypertruncated triangles

StellatedStar figure 2{3}

Truncatedt{6} = {12}

Alternatedh{6} = {3}

Crossedhexagon

A concave hexagon

A self-intersecting hexagon (star polygon)

ExtendedCentral {6} in {12}

A skew hexagon, within cube

Dissected {6}

projectionoctahedron

Complete graph

Self-crossing hexagons[edit]

There are six self-crossing hexagons with the vertex arrangement of the regular hexagon:

Self-intersecting hexagons with regular vertices

Dih2

Dih1

Dih3

Figure-eight

Center-flip

Unicursal

Fish-tail

Double-tail

Triple-tail

Hexagonal structures[edit]

Giant's Causeway closeup

From bees' honeycombs to the Giant's Causeway, hexagonal patterns are prevalent in nature due to their efficiency. In a hexagonal grid each line is as short as it can possibly be if a large area is to be filled with the fewest hexagons. This means that honeycombs require less wax to construct and gain much strength under compression.

Irregular hexagons with parallel opposite edges are called parallelogons and can also tile the plane by translation. In three dimensions, hexagonal prisms with parallel opposite faces are called parallelohedrons and these can tessellate 3-space by translation.

Hexagonal prism tessellations

Form

Hexagonal tiling

Hexagonal prismatic honeycomb

Regular

Parallelogonal

Tesselations by hexagons[edit]

Main article: Hexagonal tiling

In addition to the regular hexagon, which determines a unique tessellation of the plane, any irregular hexagon which satisfies the Conway criterion will tile the plane.

Hexagon inscribed in a conic section[edit]

Pascal's theorem (also known as the "Hexagrammum Mysticum Theorem") states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the "Pascal line" of that configuration.

Cyclic hexagon[edit]

The Lemoine hexagon is a cyclic hexagon (one inscribed in a circle) with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its symmedian point.

If the successive sides of a cyclic hexagon are a, b, c, d, e, f, then the three main diagonals intersect in a single point if and only if ace = bdf.[6]

If, for each side of a cyclic hexagon, the adjacent sides are extended to their intersection, forming a triangle exterior to the given side, then the segments connecting the circumcenters of opposite triangles are concurrent.[7]

If a hexagon has vertices on the circumcircle of an acute triangle at the six points (including three triangle vertices) where the extended altitudes of the triangle meet the circumcircle, then the area of the hexagon is twice the area of the triangle.[8]: p. 179

Hexagon tangential to a conic section[edit]

Let ABCDEF be a hexagon formed by six tangent lines of a conic section. Then Brianchon's theorem states that the three main diagonals AD, BE, and CF intersect at a single point.

In a hexagon that is tangential to a circle and that has consecutive sides a, b, c, d, e, and f,[9]

{\displaystyle a+c+e=b+d+f.}

Equilateral triangles on the sides of an arbitrary hexagon[edit]

Equilateral triangles on the sides of an arbitrary hexagon

If an equilateral triangle is constructed externally on each side of any hexagon, then the midpoints of the segments connecting the centroids of opposite triangles form another equilateral triangle.[10]: Thm. 1

Skew hexagon[edit]

A regular skew hexagon seen as edges (black) of a triangular antiprism, symmetry D3d, [2+,6], (2*3), order 12.

A skew hexagon is a skew polygon with six vertices and edges but not existing on the same plane. The interior of such a hexagon is not generally defined. A skew zig-zag hexagon has vertices alternating between two parallel planes.

A regular skew hexagon is vertex-transitive with equal edge lengths. In three dimensions it will be a zig-zag skew hexagon and can be seen in the vertices and side edges of a triangular antiprism with the same D3d, [2+,6] symmetry, order 12.

The cube and octahedron (same as triangular antiprism) have regular skew hexagons as petrie polygons.

Skew hexagons on 3-fold axes

Cube

Octahedron

Petrie polygons[edit]

The regular skew hexagon is the Petrie polygon for these higher dimensional regular, uniform and dual polyhedra and polytopes, shown in these skew orthogonal projections:

3-3 duoprism

3-3 duopyramid

5-simplex

Convex equilateral hexagon[edit]

A principal diagonal of a hexagon is a diagonal which divides the hexagon into quadrilaterals. In any convex equilateral hexagon (one with all sides equal) with common side a, there exists[11]: p.184, #286.3 a principal diagonal d1 such that

≤

{\displaystyle {\frac {d_{1}}{a}}\leq 2}

and a principal diagonal d2 such that

{\displaystyle {\frac {d_{2}}{a}}>{\sqrt {3}}.}

Polyhedra with hexagons[edit]

There is no Platonic solid made of only regular hexagons, because the hexagons tessellate, not allowing the result to "fold up". The Archimedean solids with some hexagonal faces are the truncated tetrahedron, truncated octahedron, truncated icosahedron (of soccer ball and fullerene fame), truncated cuboctahedron and the truncated icosidodecahedron. These hexagons can be considered truncated triangles, with Coxeter diagrams of the form and .

Hexagons in Archimedean solids

Tetrahedral

Octahedral

Icosahedral

truncated tetrahedron

truncated octahedron

truncated cuboctahedron

truncated icosahedron

truncated icosidodecahedron

There are other symmetry polyhedra with stretched or flattened hexagons, like these Goldberg polyhedron G(2,0):

Hexagons in Goldberg polyhedra

Tetrahedral

Octahedral

Icosahedral

Chamfered tetrahedron

Chamfered cube

Chamfered dodecahedron

There are also 9 Johnson solids with regular hexagons:

Johnson solids with hexagons

triangular cupola

elongated triangular cupola

gyroelongated triangular cupola

augmented hexagonal prism

parabiaugmented hexagonal prism

metabiaugmented hexagonal prism

triaugmented hexagonal prism

augmented truncated tetrahedron

triangular hebesphenorotunda

Prismoids with hexagons

Hexagonal prism

Hexagonal antiprism

Hexagonal pyramid

Tilings with regular hexagons

Regular

1-uniform

{6,3}

r{6,3}

rr{6,3}

tr{6,3}

2-uniform tilings

Gallery of natural and artificial hexagons[edit]

The ideal crystalline structure of graphene is a hexagonal grid.

Assembled E-ELT mirror segments

A beehive honeycomb

The scutes of a turtle's carapace

Saturn's hexagon, a hexagonal cloud pattern around the north pole of the planet

Micrograph of a snowflake

Benzene, the simplest aromatic compound with hexagonal shape.

Hexagonal order of bubbles in a foam.

Crystal structure of a molecular hexagon composed of hexagonal aromatic rings.

Naturally formed basalt columns from Giant's Causeway in Northern Ireland; large masses must cool slowly to form a polygonal fracture pattern

An aerial view of Fort Jefferson in Dry Tortugas National Park

The James Webb Space Telescope mirror is composed of 18 hexagonal segments.

In French, l'Hexagone refers to Metropolitan France for its vaguely hexagonal shape.

Hexagonal Hanksite crystal, one of many hexagonal crystal system minerals

Hexagonal barn

The Hexagon, a hexagonal theatre in Reading, Berkshire

Władysław Gliński's hexagonal chess

Pavilion in the Taiwan Botanical Gardens

Hexagonal window

What is a Hexagon? Definition, Properties, Area, Perimeter, Facts

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Hexagon – Definition, Types, Properties, Examples, FAQs

Home » Math Vocabluary » Hexagon – Definition, Types, Properties, Examples, FAQs

What is a Hexagon? Types of HexagonsClassification of hexagons based on their anglesSolved Examples on HexagonPractice Problems On HexagonFrequently Asked Questions On Hexagon

What is a Hexagon?

In geometry, a hexagon can be defined as a closed two-dimensional polygon with six sides.

Hexagon has 6 vertices and 6 angles also.

Hexa means six and gonia means angles.

Examples

Non – Examples

Hexagon in Real Life

We can find the shape of a hexagon in a honeycomb, a football, face of pencil, and floor tiles.

Hexagon around us

Begin here

2d Shapes

Identify Trapezoid, Hexagon, and Pentagon Game

Play

Types of Hexagons

Hexagonal shape is classified into several types based on the measure of sides and angles.

Classification of hexagons based on their sides

1. Regular Hexagon

When the length of all the sides and measure of all the angles are equal, it is a regular hexagon. All interior angles of a regular hexagon are 120 degrees each.

Properties of a Regular Hexagon

All the sides are equal in length.

All the interior angles measure 120°.

All the exterior angles measure 60°.

Since all angles are equal in a regular hexagon, each angle is 120o and the sum of all the interior angles is 720o.

A regular hexagon can be divided into six equilateral triangles.

The opposite sides of a regular hexagon are parallel to each other.

A regular hexagon is also a convex hexagon.

Symmetry in regular hexagon:

A regular hexagon has 6 lines of symmetry and a rotational symmetry of order 6

6 lines of symmetrySix 60° angles of rotation

2. Irregular Hexagon

In an irregular hexagon, the length of sides and measure of angles do not have the same measure.

Related Worksheets

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Classification of hexagons based on their angles

1. Convex Hexagon

For convex hexagons, all of its interior angles must be less than 180 degrees, and all the vertices are pointed outwards. Convex hexagons can be regular or irregular.

2. Concave Hexagon

For concave hexagons, at least one of its interior angles must be greater than 180° and at least one of its vertex points inwards.

Properties of Hexagon

It is two-dimensional.

It has six sides, six edges and six vertices.

The sum of the interior angles is 720 degrees.

It has nine diagonals.

Perimeter of a Hexagon

The perimeter of a hexagon is the sum of the length of all 6 sides.

Perimeter = AB + BC + CD + DE + EF + FA

In regular hexagons, all sides are equal in length. So, the perimeter of a regular hexagon is six times the length of one side.

Perimeter = a + a + a + a + a + a = 6a

Solved Examples on Hexagon

Example 1: Find the perimeter of a regular hexagon having each side measure 20 cm.

Solution:

Perimeter of regular hexagon = 6 × length of side

= 6 × 20 cm

= 120 cm

So, the perimeter of the hexagon is 120 cm.

Example 2: The perimeter of a regular hexagon is 36 cm. What is the length of its sides?

Solution:

Perimeter of regular hexagon = 6 × length of side

36 = 6 × length of side

Length of side = 366 cm = 6 cm

So, the length of its sides is 6 cm.

Example 3: Five angles of a hexagon measure 110° each. What is the measure of the sixth angle?

Solution:

Sum of the interior angles in hexagon = 720°

Sum of five angles = 5 × 110° = 550°

Sixth angle = 720° – 550° = 170°

So, the measure of the sixth angle is 170°.

Practice Problems On Hexagon

HexagonAttend this Quiz & Test your knowledge.1What is the measure of each interior angle of a regular hexagon?60°100°120°720°CorrectIncorrectCorrect answer is: 120°The sum of the interior angles of a hexagon is 720° For a regular hexagon all the sides are of the same length and all interior angles are equal. So, each interior angle= 720/6 = 120° So, the measure of the interior angle of a regular hexagon is 120°.2A plane with six sides is classified aspentagonhexagonheptagonoctagonCorrectIncorrectCorrect answer is: hexagonHexagon is a two dimensional polygon with six sides.3The hexagon is said to be convex hexagon if none of its interior angles is 90°less than 90°more than 90° and less than 180°more than 180°CorrectIncorrectCorrect answer is: more than 180°more than 180° For convex hexagons, all of its interior angles must be less than 180 degrees. For concave hexagons, at least one of its interior angles must be greater than 180°.4A regular hexagon has how many lines of symmetry 4689CorrectIncorrectCorrect answer is: 6A regular hexagon has 6 lines of symmetry. In a regular polygon the lines of symmetry are equal to the number of sides. So there will be 6 lines of symmetry for a regular hexagon.

Frequently Asked Questions On Hexagon

How many types of hexagons are there?

There are four types of hexagons. These are regular hexagons, irregular hexagons, concave hexagons, and concave hexagons.

What are the three characteristics of a hexagon?

Hexagon has 6 sides, 6 angles and 6 vertices.

What is the sum of all interior angles of a hexagon?

The sum of all interior angles of a hexagon is 720°.

How many diagonals does a hexagon have?

A hexagon has 9 diagonals.

Does a hexagon always have equal sides?

No, hexagons can have sides with different lengths. Regular hexagon has all sides are equal in length.

Are all six-sided shapes hexagon?

Yes, all six sided shapes are called hexagons.

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Hexagons Explained! The Complete Guide to Hexagons

February 5, 2021

by Anthony Persico

What is a Hexagon? - Definition, Facts, Examples, and More!Welcome to this complete guide to hexagons, where you will learn everything you need to know about this beautiful six-sided polygon!Hexagon Definition:

In mathematics and geometry, a Hexagon is defined as a polygon (a closed two-dimensional shape with straight sides) with 6 sides.Note that Hexagons have 6 sides and 6 angles.There are two types of Hexagons: Regular Hexagons and Irregular Hexagons.What is a Regular Hexagon?A regular hexagon is defined as a 6-sided polygon that is both equilateral and equiangular—meaning that all of the sides have the same length and all of the angles have the same measure.What is an Irregular Hexagon?An irregular hexagon is defined as a 6-sided polygon that is not regular—meaning that all of the sides and angles do not have the same measure.

What are the Properties of a Regular Hexagon?In Geometry, you will most often be dealing with regular hexagons. It is important to know their three main properties:All sides of a regular hexagon have equal lengths.All of the interior angles of a regular hexagon are 120° each.The total sum of the interior angles is 720°.

What Is a 3D Hexagon?

Image via www.wikipedia.org

In Geometry, a 3D Hexagon is called a Hexagonal Prism—which is a prism with hexagonal base. In the case of 3D hexagons, the hexagonal base is usually a regular hexagon.For example, a truncated octahedron can be considered a 3D Hexagon because it has a hexagonal base.Here are a few more examples of 3D Hexagons:

Convex Hexagons vs. Concave HexagonsIn Geometry, a polygon is can be convex or concave. For a hexagon to be convex, all of its interior angles must be less than 180°. For a hexagon to be concave, at least one of its interior angles must be greater than 180°. For example, a regular hexagon is also a convex polygon because all of the interior angles equal 120°, which is less than 180°.

Hexagons Degrees: Why 720°?As previously stated, the measure of each interior angle in a hexagon is 120° and the total sum of all of the interior angles is 720°. But why? Since there are 6 angles in a regular hexagon and each angle equals 120°, the total sum would be:120 + 120 + 120 + 120 + 120 + 120 = 720or120 x 6 = 720Furthermore, you can use the polygon interior sum formula to find the sum of the interior angles for any regular polygon.

By applying the polygon interior sum formula to a hexagon, you replace n with 6 (since a hexagon has 6 sides) as follows:(n - 2) x 180° ➞ (6 - 2) x 180° = 4 x 180° = 720°Hexagons in Real LifeThe hexagon is a simple yet remarkable shape that can be found everywhere and anywhere—ranging from art to architecture to nature. Here a few remarkable examples of hexagons in real life:

Hexagons in Real Life: SnowflakesDid you know that all snowflakes are hexagons? When ice crystals form, the molecules join together in a hexagonal structure. Mother Nature has determined that this type of formation is the most efficient way for snowflakes to form.

Hexagons in Real Life: HoneycombsRegular hexagons are one of only three polygons that will tesselate a plane—meaning that they can be duplicated infinitely to fill a space without any gaps. And when bees build honeycombs, they choose to use hexagons. Always!

Hexagons in Real Life: ArchitectureBees are not the only ones who understand the power and efficiency of hexagons. Ancient and modern architecture constantly utilizes this shape from floor tiles to windows to ornate ceiling designs. Hexagons are everywhere!

Hexagons in Real Life: ArtDue to their beautiful form and ability to tessellate, hexagons are constantly used in art and graphic design to create patterns, mosaics, logos, and more! In fact, many companies choose a hexagon shape for a logo because it represents strength and security.

Hexagons in Real Life: ReligionSince regular hexagons often show up in nature (like snowflakes and honeycombs) they are often included in Sacred Geometry, which assigns higher meaning and spirituality to certain shapes and proportions. In fact, some view the hexagon as the most fascinating shape in relation to the universe.

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Hexagon – Definition, Types, Properties, Examples, FAQs

Home » Math Vocabluary » Hexagon – Definition, Types, Properties, Examples, FAQs

What is a Hexagon? Types of HexagonsClassification of hexagons based on their anglesSolved Examples on HexagonPractice Problems On HexagonFrequently Asked Questions On Hexagon

What is a Hexagon?

In geometry, a hexagon can be defined as a closed two-dimensional polygon with six sides.

Hexagon has 6 vertices and 6 angles also.

Hexa means six and gonia means angles.

Examples

Non – Examples

Hexagon in Real Life

We can find the shape of a hexagon in a honeycomb, a football, face of pencil, and floor tiles.

Hexagon around us

Begin here

2d Shapes

Identify Trapezoid, Hexagon, and Pentagon Game

Play

Types of Hexagons

Hexagonal shape is classified into several types based on the measure of sides and angles.

Classification of hexagons based on their sides

1. Regular Hexagon

When the length of all the sides and measure of all the angles are equal, it is a regular hexagon. All interior angles of a regular hexagon are 120 degrees each.

Properties of a Regular Hexagon

All the sides are equal in length.

All the interior angles measure 120°.

All the exterior angles measure 60°.

Since all angles are equal in a regular hexagon, each angle is 120o and the sum of all the interior angles is 720o.

A regular hexagon can be divided into six equilateral triangles.

The opposite sides of a regular hexagon are parallel to each other.

A regular hexagon is also a convex hexagon.

Symmetry in regular hexagon:

A regular hexagon has 6 lines of symmetry and a rotational symmetry of order 6

6 lines of symmetrySix 60° angles of rotation

2. Irregular Hexagon

In an irregular hexagon, the length of sides and measure of angles do not have the same measure.

Related Worksheets

View

Classification of hexagons based on their angles

1. Convex Hexagon

For convex hexagons, all of its interior angles must be less than 180 degrees, and all the vertices are pointed outwards. Convex hexagons can be regular or irregular.

2. Concave Hexagon

For concave hexagons, at least one of its interior angles must be greater than 180° and at least one of its vertex points inwards.

Properties of Hexagon

It is two-dimensional.

It has six sides, six edges and six vertices.

The sum of the interior angles is 720 degrees.

It has nine diagonals.

Perimeter of a Hexagon

The perimeter of a hexagon is the sum of the length of all 6 sides.

Perimeter = AB + BC + CD + DE + EF + FA

In regular hexagons, all sides are equal in length. So, the perimeter of a regular hexagon is six times the length of one side.

Perimeter = a + a + a + a + a + a = 6a

Solved Examples on Hexagon

Example 1: Find the perimeter of a regular hexagon having each side measure 20 cm.

Solution:

Perimeter of regular hexagon = 6 × length of side

= 6 × 20 cm

= 120 cm

So, the perimeter of the hexagon is 120 cm.

Example 2: The perimeter of a regular hexagon is 36 cm. What is the length of its sides?

Solution:

Perimeter of regular hexagon = 6 × length of side

36 = 6 × length of side

Length of side = 366 cm = 6 cm

So, the length of its sides is 6 cm.

Example 3: Five angles of a hexagon measure 110° each. What is the measure of the sixth angle?

Solution:

Sum of the interior angles in hexagon = 720°

Sum of five angles = 5 × 110° = 550°

Sixth angle = 720° – 550° = 170°

So, the measure of the sixth angle is 170°.

Practice Problems On Hexagon

Frequently Asked Questions On Hexagon

How many types of hexagons are there?

There are four types of hexagons. These are regular hexagons, irregular hexagons, concave hexagons, and concave hexagons.

What are the three characteristics of a hexagon?

Hexagon has 6 sides, 6 angles and 6 vertices.

What is the sum of all interior angles of a hexagon?

The sum of all interior angles of a hexagon is 720°.

How many diagonals does a hexagon have?

A hexagon has 9 diagonals.

Does a hexagon always have equal sides?

No, hexagons can have sides with different lengths. Regular hexagon has all sides are equal in length.

Are all six-sided shapes hexagon?

Yes, all six sided shapes are called hexagons.

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Hexagons Explained! The Complete Guide to Hexagons — Mashup Math

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February 5, 2021

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What Is a 3D Hexagon?

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Hexagon – Definition, Types, Properties, Examples, FAQs

Home » Math Vocabluary » Hexagon – Definition, Types, Properties, Examples, FAQs

What is a Hexagon? Types of HexagonsClassification of hexagons based on their anglesSolved Examples on HexagonPractice Problems On HexagonFrequently Asked Questions On Hexagon

What is a Hexagon?

In geometry, a hexagon can be defined as a closed two-dimensional polygon with six sides.

Hexagon has 6 vertices and 6 angles also.

Hexa means six and gonia means angles.

Examples

Non – Examples

Hexagon in Real Life

We can find the shape of a hexagon in a honeycomb, a football, face of pencil, and floor tiles.

Hexagon around us

Begin here

2d Shapes

Identify Trapezoid, Hexagon, and Pentagon Game

Play

Types of Hexagons

Hexagonal shape is classified into several types based on the measure of sides and angles.

Classification of hexagons based on their sides

1. Regular Hexagon

When the length of all the sides and measure of all the angles are equal, it is a regular hexagon. All interior angles of a regular hexagon are 120 degrees each.

Properties of a Regular Hexagon

All the sides are equal in length.

All the interior angles measure 120°.

All the exterior angles measure 60°.

Since all angles are equal in a regular hexagon, each angle is 120o and the sum of all the interior angles is 720o.

A regular hexagon can be divided into six equilateral triangles.

The opposite sides of a regular hexagon are parallel to each other.

A regular hexagon is also a convex hexagon.

Symmetry in regular hexagon:

A regular hexagon has 6 lines of symmetry and a rotational symmetry of order 6

6 lines of symmetrySix 60° angles of rotation

2. Irregular Hexagon

In an irregular hexagon, the length of sides and measure of angles do not have the same measure.

Related Worksheets

View

Classification of hexagons based on their angles

1. Convex Hexagon

For convex hexagons, all of its interior angles must be less than 180 degrees, and all the vertices are pointed outwards. Convex hexagons can be regular or irregular.

2. Concave Hexagon

For concave hexagons, at least one of its interior angles must be greater than 180° and at least one of its vertex points inwards.

Properties of Hexagon

It is two-dimensional.

It has six sides, six edges and six vertices.

The sum of the interior angles is 720 degrees.

It has nine diagonals.

Perimeter of a Hexagon

The perimeter of a hexagon is the sum of the length of all 6 sides.

Perimeter = AB + BC + CD + DE + EF + FA

In regular hexagons, all sides are equal in length. So, the perimeter of a regular hexagon is six times the length of one side.

Perimeter = a + a + a + a + a + a = 6a

Solved Examples on Hexagon

Example 1: Find the perimeter of a regular hexagon having each side measure 20 cm.

Solution:

Perimeter of regular hexagon = 6 × length of side

= 6 × 20 cm

= 120 cm

So, the perimeter of the hexagon is 120 cm.

Example 2: The perimeter of a regular hexagon is 36 cm. What is the length of its sides?

Solution:

Perimeter of regular hexagon = 6 × length of side

36 = 6 × length of side

Length of side = 366 cm = 6 cm

So, the length of its sides is 6 cm.

Example 3: Five angles of a hexagon measure 110° each. What is the measure of the sixth angle?

Solution:

Sum of the interior angles in hexagon = 720°

Sum of five angles = 5 × 110° = 550°

Sixth angle = 720° – 550° = 170°

So, the measure of the sixth angle is 170°.

Practice Problems On Hexagon

Frequently Asked Questions On Hexagon

How many types of hexagons are there?

There are four types of hexagons. These are regular hexagons, irregular hexagons, concave hexagons, and concave hexagons.

What are the three characteristics of a hexagon?

Hexagon has 6 sides, 6 angles and 6 vertices.

What is the sum of all interior angles of a hexagon?

The sum of all interior angles of a hexagon is 720°.

How many diagonals does a hexagon have?

A hexagon has 9 diagonals.

Does a hexagon always have equal sides?

No, hexagons can have sides with different lengths. Regular hexagon has all sides are equal in length.

Are all six-sided shapes hexagon?

Yes, all six sided shapes are called hexagons.

Frequency Table in Math – Definition, FAQs, Examples

Straight Angle – Definition With Examples

What Are Rectilinear Figures? Definition, Types, Examples, FAQs

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hexagon 在英语-中文（简体）词典中的翻译

hexagonnoun [ C ] uk

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/ˈhek.sə.ɡən/ us

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/-ɡɑːn/

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a shape with six straight sides

六边形，六角形

(hexagon在剑桥英语-中文（简体）词典的翻译 © Cambridge University Press)

hexagon的例句

hexagon

Furthermore, we demonstrate computationally that there are other stable patterns composed of both rolls and hexagons.

来自 Cambridge English Corpus

Finally, between (iii) and (iv), the mixed modes become false hexagons in a transcritical bifurcation at the hexagons.

来自 Cambridge English Corpus

I believe that anyone who thinks that the restricted betterness relation is strongly separable across people should also accept the hexagon condition in such cases.

来自 Cambridge English Corpus

We, therefore, were able to precisely control the contrast and mean luminance of the center hexagon.

来自 Cambridge English Corpus

The hexagons of any layer below the top nest within the hexagon of the higher-level centre they depend on.

来自 Cambridge English Corpus

We also examined whether the proximity of the surrounding luminance influenced the amplitude and implicit time of the center hexagon's response.

来自 Cambridge English Corpus

The anger emotion was then joined to the happiness emotion to produce the emotion hexagon.

来自 Cambridge English Corpus

The last sorts of polygon we mention here are equi-angled hexagons.

来自 Cambridge English Corpus

示例中的观点不代表剑桥词典编辑、剑桥大学出版社和其许可证颁发者的观点。

hexagon的翻译

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六邊形，六角形…

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hexágono…

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Hexagon Shape - Sides of Hexagon | Regular Hexagon | Angles of Hexagon

gon Shape - Sides of Hexagon | Regular Hexagon | Angles of HexagonAlready booked a tutor?Book A FREE ClassLearn Hexagon with tutors mapped to your child's learning needs.30-DAY PROMISE | GET 100% MONEY BACK*Book A FREE Class*T&C ApplyLearnPracticeDownload

Hexagon

A hexagon is defined as a closed 2D shape that is made up of six straight lines. It is a 6 sided polygon which means it has six sides, six vertices, and six interior angles. Let us learn about hexagon shape, the internal angles of hexagon, the properties of hexagon, the diagonals of a hexagon, regular hexagon, and hexagon examples on this page.

What is Hexagon?

Regular Hexagon

Hexagon Sides

Angles of Hexagon

Diagonals of a Hexagon

Types of Hexagon

Properties of Hexagon

FAQs on Hexagon Shape

What is Hexagon?

Hexagon is a two-dimensional geometrical shape that is made of six sides and six angles. Some real-life examples of the hexagon shape are a hexagonal floor tile, pencil cross-section, clock, honeycomb, etc. A hexagon can be either a regular hexagon which has 6 equal sides and 6 equal interior angles, or an irregular hexagon which has 6 sides of different lengths and 6 angles of different measure.

Hexagon Definition

The hexagon definition states that a hexagon is a 6 sided polygon and the name is derived from a Greek word where 'hex' means six, and 'gonia' means corners. This means it has 6 sides, 6 corners, and 6 interior angles.

Regular Hexagon

A regular hexagon is defined as a closed 2D shape made up of six equal sides and six equal angles. Each angle of the regular hexagon measures 120°. The sum of all the interior angles is 120 × 6 = 720°. When it comes to the exterior angles, we know that the sum of exterior angles of any polygon is always 360°. There are 6 exterior angles in a hexagon. So, each of the exterior angles in a regular hexagon is equal to 60°.

Irregular Hexagon

In an irregular hexagon, the 6 sides are of different lengths and the 6 interior angles are of different measures.

Some of the properties that are common to both irregular and regular hexagons are given below:

There are 6 sides, 6 interior angles, and 6 vertices in both.

The sum of all 6 interior angles is always 720°.

The sum of all 6 exterior angles is always 360°.

Hexagon Sides

There are six sides in a hexagon, as shown in the figure given above. All are straight edges and form a closed shape. In a regular hexagon, we have six equal sides, while in an irregular hexagon, at least two of the sides of a hexagon are different in measure. If we take the sum of all six sides, we will get the perimeter of the hexagon.

In a regular hexagon, if we know the value of the perimeter, then the length of each side can be calculated as "Perimeter ÷ 6". For example, if the perimeter of a regular hexagon is 72 units, then the length of each of the hexagon sides is 72 ÷ 6 = 12 units.

Angles of Hexagon

There are six internal angles and six exterior angles in a hexagon. The sum of all six angles of hexagon is 720°, while the sum of its exterior angles is 360°. Observe the properties related to hexagon angles listed below:

Angles of a Regular Hexagon

The measurement of each interior angle in a regular hexagon is 720° ÷ 6 = 120°.

The measurement of each exterior angle of a regular hexagon is 360° ÷ 6 = 60°.

Angles of an Irregular Hexagon

At least two of the angles are of different measurements in an irregular hexagon.

Diagonals of a Hexagon

A diagonal is a segment of a line, that connects any two non-adjacent vertices of a polygon. The number of diagonals of a polygon is given by n(n-3)/2, where 'n' is the number of sides of a polygon. The number of diagonals in a hexagon is given by, 6 (6 - 3) / 2 = 6(3)/2, which is 9. Out of the 9 diagonals, 3 of them pass through the center of the hexagon.

Types of Hexagon

Hexagons can be classified based on their side lengths and internal angles. Considering the sides and angles of a hexagon, the types of the hexagon are:

Regular Hexagon: This is a hexagon that has equal sides and angles. All the internal angles of a regular hexagon are 120° each. The exterior angles measure 60° each. The sum of the interior angles of a regular hexagon is 6 times 120°, which is equal to 720°. The sum of the exterior angles is equal to 6 times 60°, which is equal to 360°.

Irregular Hexagon: This is a hexagon that has sides and angles of different measurements. All the internal angles are not equal to 120°. But, the sum of all interior angles is the same, i.e., 720°.

Convex Hexagon: This is a hexagon in which all the interior angles measure less than 180°. Convex hexagons can be regular or irregular, which means they can have equal or unequal side lengths and angles. All the vertices of the convex hexagon are pointed outwards.

Concave Hexagon: This is a hexagon in which at least one of the interior angles is greater than 180°. There is at least one vertex that points inwards.

Properties of Hexagon

A hexagon is a flat two-dimensional six sided shape. It may or may not have equal sides and angles. Based on these facts, the important properties of a hexagon are as follows:

It has six sides, six edges, and six vertices.

All the side lengths are equal or unequal in measurement.

All the internal angles are equal to 120° each in a regular hexagon.

The sum of the internal angles is always equal to 720°.

All the external angles are equal to 60° each in a regular hexagon.

The sum of the exterior angles is equal to 360°.

The number of diagonals that can be drawn in a hexagon is 9.

A regular hexagon is also a convex hexagon since all its internal angles are less than 180°.

It can be split into six equilateral triangles.

It is symmetrical as each of its side lengths is equal.

The opposite sides of a regular hexagon are always parallel to each other.

The area of a regular hexagon is 3√3a2/2 square units, where a is the side length.

The hexagon's perimeter can be found by adding the lengths of all six sides.

☛ Related Topics

Check out some interesting articles related to hexagon shape in math.

Pentagon - 5 sided polygon

Heptagon - 7 sided polygon

Octagon - 8 sided polygon

Decagon - 10 sided polygon

Hexagon Examples

Example 1: State true or false:

a.) All the internal angles in a regular hexagon are equal to 120° each.

b.) A hexagon can be split into 4 equilateral triangles.

Solution:

a.) True, all the internal angles in a regular hexagon are equal to 120° each.

b.) False, a hexagon can be split into 6 equilateral triangles.

Example 2: Fill in the blanks:

a.) _ is a 6 sided polygon.

b.) All the external angles are equal to __ each in a regular hexagon.

Solution:

a.) Hexagon is a 6 sided polygon.

b.) All the external angles are equal to 60° each in a regular hexagon.

Example 3: What is the length of each side of a regular hexagon, if its perimeter is equal to 108 units?

Solution:

Given, the perimeter = 108 units.

Since it is a regular hexagon, all its sides are of equal length. To find the length of each side, we just need to divide the perimeter by 6.

⇒ Perimeter ÷ 6

⇒ 108 ÷ 6

= 18 units

Therefore, the length of each side of the hexagon is 18 units.

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FAQs on Hexagon

What is a Hexagon Shape?

A hexagon is a two-dimensional flat shape that has six angles, six edges, and six vertices. It can have equal or unequal sides and interior angles. It is a 6-sided polygon classified into two main types - regular and irregular hexagon.

What are the Angles of a Hexagon?

A hexagon has six angles and the sum of all six interior angles is 720°. In a regular hexagon, each interior angle measures 120°.

What is a Regular Hexagon?

A regular hexagon is defined as a special type of hexagon that has all sides equal. All six angles in a regular hexagon are also equal.

How many Sides does a Hexagon have?

A hexagon has six sides. All six sides join together and form a closed shape known as a hexagon. In a regular hexagon, all six sides are of equal lengths, while in an irregular hexagon, there is no definite relationship between the sides as they are different in measure.

What is the Perimeter of a Hexagon?

The perimeter of a hexagon is the sum of its boundary. It is the sum of all six sides. In the case of a regular hexagon, the formula to calculate its perimeter is 6 × side length.

What is an Irregular Hexagon?

An irregular hexagon has at least one unequal side and angle when compared to the other sides and angles. There is no definite measurement of each of the angles, but the sum of all 6 interior angles is always 720°, and the sum of all 6 exterior angles is 360°.

What are the Three Attributes of a Hexagon?

The three attributes of a hexagon are:

It has 6 sides

It has 6 angles

It has 6 vertices

Does a Hexagon Always Have Equal Sides?

No, a hexagon may not necessarily have all sides equal. It can have sides of variable lengths too. The hexagon having equal sides is called a regular hexagon and the one with different sides is called an irregular hexagon.

How are Hexagons Classified?

A hexagon is classified based on the side lengths and angles. Based on this, hexagons are classified into the following types:

Regular hexagons that have equal side lengths and angles.

Irregular hexagons that have unequal side lengths and angles.

Convex hexagons in which all the interior angles are less than 180°

Concave hexagons in which at least one of the interior angles is greater than 180°.

What is the Sum of Interior Angles of a Hexagon?

In a hexagon, the sum of all 6 interior angles is always 720º. The sum of interior angles of a polygon is calculated using the formula, (n-2) × 180°, where 'n' is the number of sides of the polygon. Since a hexagon has 6 sides, taking 'n' as 6 we get (6-2) × 180°, which gives 720°.

How many Diagonals does a Regular Hexagon have?

The formula to calculate the number of diagonals of a polygon is n(n-3)/2, where 'n' is the number of sides of the polygon. After substituting the value of n = 6 in the formula, we get 6(6-3)/2, which is equal to 9. Therefore, a regular hexagon has 9 diagonals.

How many Lines of Symmetry are there in a Regular Hexagon?

For all regular polygons, the number of lines of symmetry is equal to the number of sides. Thus, for a regular hexagon, there are six lines of symmetry.

How to Find the Area of a Hexagon?

We can determine the area of a hexagon by identifying the length of the side of the hexagon. To find the area of a regular hexagon we use the formula, A = (3√3 S2)/2 square units, where S is the length of one side.

What is the Formula to Calculate the Perimeter of a Hexagon?

The formula to calculate the regular hexagon perimeter is 6a, where 'a' is the side length of the hexagon. In the case of an irregular hexagon, we add the side lengths. Mathematically, it can be expressed as,

Perimeter of hexagon = sum of all the 6 sides

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Hexagon

A hexagon is a 6-sided polygon (a flat shape with straight sides):

images/area-coords.js?mode=6

Soap bubbles tend to form hexagons

when they join up.

Honeycomb has hexagons too!

Regular or Irregular

When all angles are equal and all sides are equal it is regular, otherwise it is irregular:

Regular Hexagon

Irregular Hexagons

Concave or Convex

A convex hexagon has no angles pointing inwards. More precisely, no internal angles can be more than 180°.

When any internal angle is greater than 180° it is concave. (Think: concave has a "cave" in it)

Convex Hexagon

Concave Hexagon

Is it a Hexagon?

No curved sides. And the shape must also be closed (all the lines connect up):

Hexagon

(straight sides)

Not a Hexagon

(has a curve)

Not a Hexagon

(open, not closed)

Properties

A regular hexagon has:

Interior Angles of 120°

Exterior Angles of 60°

Area = (1.5√3) × s2 , or approximately 2.5980762 × s2 (where s=side length)

Radius equals side length

The radius is the side length.

It is also made of 6 regular triangles!

Any hexagon has:

Sum of Interior Angles of 720°

9 diagonals

More Images

Hexagonal nuts and bolts are easy to grip with a wrench,

which can be re-positioned every 60° if needed.

There is a huge hexagon on Saturn,

it is wider than Earth.

Another image of the hexagon on Saturn.

Snowflakes have hexagonal patterns, like this beautiful image from NASA.

Photograph by NASA / Alexey Kljatov.

Also a snowflake!

Photograph by NASA / Alexey Kljatov.

4891, 629, 676, 758, 782, 3261, 3260, 633, 6069, 2884

Geometry Index

All About Hexagons - Definition, Examples, Formulas - DoodleLearning

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All About Hexagons

Discover the ins and outs of this versatile six-sided shape.

AuthorTaylor Hartley

Expert ReviewerJill Padfield

Published: August 24, 2023

All About Hexagons

Discover the ins and outs of this versatile six-sided shape.

AuthorTaylor Hartley

Expert ReviewerJill Padfield

Published: August 24, 2023

All About Hexagons

Discover the ins and outs of this versatile six-sided shape.

AuthorTaylor Hartley

Expert ReviewerJill Padfield

Published: August 24, 2023

Key takeaways

A hexagon has six sides – They are two-dimensional closed figures, and they can be oddly shaped, so long as their sides add up to six.There are different kinds of hexagons – Regular hexagons have equal sides and equal angles, while irregular hexagons can bend the rules a little. The hexagon shape is popular in nature – Beehives, insect eyes and more feature hexagons because of their ability to fit together seamlessly.

Table of contents

Key takeaways

What is a hexagon?

Must-have properties of a hexagon

What can we measure about a hexagon?

Let’s practice together!

Practice problems

FAQs

Have you ever stared at a beehive and thought, “Wow, those shapes are really cool!”? What you’ve actually been looking at are natural hexagons – six-sided polygons that fit together perfectly! So what is a hexagon, and how can we measure its perimeter, area, and more? Let’s take a closer look at this unique shape

What is a hexagon?

A hexagon is a six-sided polygon with six interior angles. There are two different types of hexagons – regular and irregular hexagons. To be considered “regular,” a hexagon must have six equal sides and six angles that each measure 120°. So where does the word “hexagon” come from? Well, the prefix “hexa” means six, and “gonia” means angles. Put that together, and you get “six angles,” a requirement for a shape to be called a hexagon!

Hexagons in the real world

Hexagons are some of the most naturally occurring shapes in the world! That’s because six sides tend to fit together perfectly to make a strong, dynamic shape or design. You’ll see hexagons when you study beehives, insect eyes, and the center of snowflakes. Ever kicked a soccer ball around? The black and white designs that make up the outside of the ball are hexagons! They are also a popular choice for tile floors, given they fit so neatly together.

Must-have properties of a hexagon

Fortunately, hexagons don’t have as many rules to follow as, say, a square, a rectangle, or a triangle. Basically, a hexagon has: 6 sides 6 angles6 vertices See the major pattern here? Six, six, six! However, there are different types of hexagons, and each of them have their own set of rules. Let’s take a look at these different types together.

Regular hexagons

Regular hexagons follow more rules than other types of hexagons. In order to be considered a regular hexagon, the shape must have: 6 equal sides6 vertices6 interior angles that each measure 120° If you’re calculating the perimeter of a regular hexagon, you simply need to multiply the length of one side by 6 to get the perimeter of the entire shape. Some other fun facts about a regular hexagon: All exterior angles measure 60°The shape can be divided into six equilateral trianglesThe opposite sides of a regular hexagon are parallel to each other

Convex hexagons

Every regular hexagon will also be a convex hexagon. When something is convex, it means that it only has interior angles. In a regular hexagon, all angles must be interior, so a regular hexagon is also a convex hexagon.

Irregular hexagons

When we think about the word irregular, we think about “strange” or a little whacky, right? Think about irregular hexagons in the same way. They still have to have six sides, and they must be a closed shape, but outside of that, they get to break the rules.

Concave hexagons

The word “concave” literally has the word “cave” in it, right? In a concave hexagon, at least one of the angles will be an exterior angle, which may make the shape look like it has a mouth! Concave hexagons still have to have – you guessed it – six sides! But two of those sides will likely point inward rather than creating a smooth, round perimeter.

What can we measure about a hexagon?

Shapes like circles and squares have specific formulas for measuring area and perimeter. There is a formula to measure the area of a hexagon, but you likely will not need it until much later in your academic career. We are going to focus on the measurement you’ll likely need to calculate once you start working with hexagons: perimeter.

The perimeter of a hexagon

You probably know that the perimeter of any shape is the measure of its exterior sides. To calculate the perimeter of a hexagon, simply add up each of the shape’s sides. For a regular hexagon, you can do this pretty easily. Simply use the measurement of one side, then multiply that number by 6. For an irregular hexagon, you will want to add up each of the sides on your own.

Let’s practice together!

1. Identify the regular hexagon.

The correct answer is A. A is the only hexagon that has 6 equal sides and 6 equal angles. Both B and C are irregular hexagons.

2. True or false: Figure C in question #1 is a convex hexagon.

The answer is FALSE. Figure C is actually a concave hexagon, while Figures A and B are convex hexagons. See how Figure C has a piece that looks like a cave or a mouth? That’s how you can tell!

3. What is the perimeter of a regular hexagon with one side that measures 7 inches?

Since we’re calculating the perimeter of a regular hexagon, we know that we can multiply one side by 6 to get the perimeter. So, 7 × 6 = 42 inches.

Ready to give it a go?

It’s time to put your knowledge of hexagons to the ultimate test! Work through the following problems on your own. Feel free to look back at the practice problems above if you get stuck, or use the guide to refresh your memory. And remember, don’t get discouraged if you run into a roadblock! Practice is the best way to learn something new.

Practice problems

Click to reveal the answer.

1. What is the perimeter of a regular hexagon with one side that measures 10 inches?

The answer is 60 inches.

2. What is the perimeter of this irregular hexagon?

The answer is 30 inches.

3. True or false: A convex hexagon is always a regular hexagon.

The answer is false.

4. Calculate the perimeter of a regular hexagon with one side that measures 5 inches.

The answer is 30 inches.

5. True or false: A concave hexagon is never a regular hexagon.

The answer is true.

Parent Guide

The answer is line 60 inches.How did we get here? Since this is a regular hexagon, we can assume that all sides are equal. Multiply 10 by 6, or add 10 together 6 times to get 60 inches.

The answer is 30 inches. How did we get here? For an irregular hexagon, you have to add all the sides together. We can make this a bit quicker using multiplication. 4 sides equal 6 inches, so 6 × 4 = 24. 2 sides equal 3 inches, and 3 × 2 = 6. Then, we add 24 inches and 6 inches together to get 30 inches.

The answer is false. How did we get here?It’s true that all regular hexagons are convex, but not ALL convex hexagons are regular. You can have irregular convex hexagons. That means the answer must be false.

The answer is 30 inches. How did we get here?Since this is a regular hexagon, we can assume that all sides are equal. Multiply 5 by 6, or add 5 together 6 times to get 30 inches.

The answer is true. How did we get here?Since concave hexagons include at least one exterior angle, they cannot be regular hexagons, which must have 6 interior angles that measure 120°.

Turn math into an adventure when you sign up for DoodleMath.Click here to get started for free!

FAQs about math strategies for kids

We understand that diving into new information can sometimes be overwhelming, and questions often arise. That’s why we’ve meticulously crafted these FAQs, based on real questions from students and parents. We’ve got you covered!

Are all 6-sided shapes called hexagons?

No, not all six-sided shapes are called hexagons. A hexagon is a specific type of six-sided shape with straight sides. It has six angles and six sides that are all equal in length. Each side of a hexagon connects to two other sides, and it has a total of six corners, which are called vertices.

Do all hexagons have equal sides?

No, not all hexagons have equal sides. A hexagon is a polygon with six sides, but its sides can have different lengths. A hexagon with equal sides is called a regular hexagon. In a regular hexagon, all six sides are the same length, and all six angles are equal.

Why is the hexagon the strongest shape?

The hexagon is often considered a strong shape because it distributes forces or weight evenly across its sides. This means that when you push or pull on a hexagon, the pressure or force is spread out more evenly compared to other shapes.

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Taylor HartleyTaylor Hartley is an author and an English teacher based in Charlotte, North Carolina. When she's not writing, you can find her on the rowing machine or lost in a good novel.

Jill PadfieldJill Padfield has 7 years of experience teaching high school mathematics, ranging from Alegra 1 to AP Calculas. She is currently working as a Business Analyst, working to improve services for Veterans while earning a masters degree in business administration.

Taylor HartleyTaylor Hartley is an author and an English teacher based in Charlotte, North Carolina. When she's not writing, you can find her on the rowing machine or lost in a good novel.

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Hexagon -- from Wolfram MathWorld

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Hexagon

Download Wolfram Notebook

A hexagon is a six-sided polygon. Several special types of hexagons are illustrated above. In particular, a hexagon with vertices equally

spaced around a circle and with all sides the same length is a regular

polygon known as a regular hexagon.

Given an arbitrary hexagon, take each three consecutive vertices, and mark the fourth point of the parallelogram sharing these three

vertices. Taking alternate points then gives two congruent triangles, as illustrated

above (Wells 1991).

Given an arbitrary hexagon, connecting the centroids of each consecutive three sides gives a hexagon with equal and parallel sides known as the centroid

hexagon (Wells 1991).

See alsoCentroid Hexagon, Octagon, Pentagon, Polygon, Regular

Hexagon Explore this

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ReferencesWells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,

pp. 53-54, 1991.

Cite this as:

Weisstein, Eric W. "Hexagon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Hexagon.html

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Hexagon – Definition, Geometry, Applications, and Examples

JUMP TO TOPICDefinitionHistorical SignificanceHoneycombsCrystallographyNASA’s Hexagon ExperimentAncient ArchitectureArt and DesignScared Geometry Scientific SignificanceHexagon’s GeometryHexagon TypesRegular HexagonIrregular HexagonConvex HexagonConcave HexagonRelevant FormulaePerimeterInterior AngleExterior AngleAreaDiagonalsApothem and RadiusUsing the Apothem or Radius for AreaApplicationsMathematics and GeometryArchitecture and ConstructionManufacturing and EngineeringComputer Science and Data RepresentationNature and BiologyCommunication and NetworkingExerciseExample 1SolutionExample 2SolutionExample 3SolutionExample 4SolutionExample 5SolutionExample 6SolutionExample 7SolutionExample 8Solution

A hexagon represents a perfect balance between intricacy and simplicity, casting a unique spotlight on our mathematical and natural world. This six-sided polygon, the hexagon, might appear to be just another ordinary shape, a cursory glance at your elementary school geometry lessons. However, a deeper examination reveals an intricate pattern and underlying beauty that extends far beyond elementary education into the advanced realms of chemistry, art, architecture, and even the cosmos. Hexagonal patterns are omnipresent, whether exploring the compact honeycomb structures formed by diligent bees, the snowflake’s microscopic symmetry, or the mysterious Giant’s Causeway in Northern Ireland.Read moreHalfplane: Definition, Detailed Examples, and MeaningThis extensive article aims to guide you through the remarkable world of hexagons, revealing their hidden elegance and emphasizing their relevance in our daily lives and the universe beyond.DefinitionA hexagon is a geometric figure, a polygon, that has six sides and six vertices. The term originates from the Greek words “hex,” meaning six, and “gonia,” meaning angles. The nature of a hexagon can vary depending on its attributes. If all of its sides and angles are equal, it’s known as a regular hexagon, exhibiting perfect symmetry. Below is the generic diagram for a Hexagon having 6 sides.Read moreHow to Find the Volume of the Composite Solid?Figure-1: Generic hexagon.Regular hexagons have interior angles of 120 degrees each, exterior angles of 60 degrees each, and the ability to tile a plane without any gaps, also known as tessellation. Conversely, an irregular hexagon has sides and angles that do not necessarily share the same measurements, resulting in a lack of symmetry. Hexagons are prevalent in various aspects of life, from natural phenomena to human-made designs and systems, due to their unique geometric properties.Historical SignificanceHexagons have a long and rich history in both natural phenomena and human-made structures and symbols, underpinning their historical significance in various cultures and scientific disciplines.HoneycombsThe hexagonal shape is commonly found in nature, especially in honeycombs built by bees. The honeycomb’s hexagonal structure allows for maximum efficiency in storing honey and maintaining structural stability. The use of hexagons in honeycombs has inspired architects and engineers to incorporate similar principles in their designs.CrystallographyHexagonal structures are prevalent in crystallography. Many minerals and crystals, such as quartz and snowflakes, exhibit hexagonal symmetry in their atomic arrangement. The study of crystallography has been instrumental in understanding the nature of matter and has contributed significantly to fields like materials science and chemistry.NASA’s Hexagon ExperimentIn recent history, hexagons gained attention through NASA’s study of Saturn’s north polar storm system, known as the “hexagon.” The Cassini spacecraft, launched in 1997, captured images and data that revealed a persistent hexagonal cloud pattern at Saturn’s north pole. This discovery has fascinated scientists and sparked an interest in understanding atmospheric dynamics and the potential formation of hexagonal patterns in natural systems.Ancient ArchitectureThe use of hexagons in human structures can be traced back to ancient architecture. The most famous ancient example is perhaps the Roman Pantheon, a former Roman temple with a hexagonal pattern embedded in the architecture of the massive dome. This pattern was likely chosen for both aesthetic and structural reasons, as the tessellation of hexagons could evenly distribute the weight of the dome.Art and DesignIn art and design, hexagonal patterns have been utilized across different cultures and historical periods. For example, the Islamic geometric pattern often includes hexagonal tessellations for their intricate design and symbolism. Additionally, the Chinese cultural significance of the number six, often associated with smoothness and success, makes hexagonal patterns a popular motif in various artifacts and traditional designs.Scared Geometry Hexagons also play a significant role in sacred geometry, a form of symbolism and design found in spiritual and religious contexts. The Star of David, a hexagram, is one of the most recognizable symbols of Judaism. In Christianity, the “Seal of Solomon,” another hexagonal pattern, has historical significance. The hexagon represents harmony and balance in some interpretations of sacred geometry.Scientific SignificanceFrom a scientific perspective, the historical significance of hexagons is quite astounding. In the natural world, hexagons are evident in the honeycomb structures made by bees. This shape is used because it is the most resource-efficient: With the least amount of wax, it enables the bees to store the most honey possible. The Giant’s Causeway, a natural rock formation in Northern Ireland, also features predominantly hexagonal columns. The cooling and cracking of hot volcanic basalt form these.On a larger scale, hexagons appear in atmospheric patterns on planets in our solar system. The most famous of these is the persistent hexagonal cloud pattern at Saturn’s North Pole, discovered by the Voyager missions and further investigated by the Cassini mission.Hexagon’s GeometryA hexagon, from a geometric perspective, is a six-sided polygon with six angles. The properties of a hexagon can differ considerably depending on whether it is a regular or an irregular hexagon.Let’s start with a regular hexagon, which is a special type of hexagon where all sides and angles are congruent.Each internal angle is 120 degrees. This is because the sum of the internal angles of any polygon is (n-2) × 180 degrees, where n is the number of sides. For a hexagon, this gives us (6-2) × 180 = 720 degrees. Dividing this by 6 (since a regular hexagon has 6 equal angles), each internal angle measures 120 degrees.Each external angle of a regular hexagon is 60 degrees. This comes from the fact that the sum of the internal angle and the adjacent external angle on a straight line is 180 degrees. So, 180 – 120 (the internal angle) = 60 degrees.The perimeter of a regular hexagon is simply six times the length of one side, given that all sides are of equal length.The formula can be used to get the area (A) of a regular hexagon. $A = \frac{3 \times \sqrt{3}}{2} \times s^2 $, where s is the length of a side. This equation is the result of the division of a regular hexagon into six equilateral triangles.An irregular hexagon, on the other hand, has sides and angles that are not necessarily congruent. Because of the variability of an irregular hexagon’s dimensions, there are no simple formulas to calculate the area or perimeter. Typically, to find these values, you’d divide the hexagon into triangles or other shapes, find the areas of these, and then sum them.Hexagon TypesHexagons can be categorized into different types based on their properties and characteristics. Here are the main types of hexagons.Regular HexagonA regular hexagon is a hexagon where all six sides and all six angles are equal. It has rotational symmetry of order six, meaning that it can be rotated by multiples of 60 degrees and still maintain the same shape. Regular hexagons are highly symmetrical and have consistent side lengths and interior angles. Below is the diagram for an irregular Hexagon. Figure-2: Irregular hexagon.Irregular HexagonAn irregular hexagon is a hexagon that does not have equal side lengths or equal interior angles. Its sides and angles can have different measures, and it lacks the rotational symmetry found in regular hexagons. Irregular hexagons can have a variety of shapes and sizes. Below we present the geometric shape of an irregular Hexagon. Figure-3: Irregular hexagon.Convex HexagonA convex hexagon is a hexagon in which no interior angle is greater than 180 degrees. All the sides of a convex hexagon point outwards, giving it a “bowed out” appearance. Convex hexagons are often the focus of geometric studies due to their simplicity and well-defined properties. Below we present the geometric diagram of an irregular convex Hexagon.Figure-4: Irregular convex hexagon.Concave HexagonA concave hexagon is a hexagon whose inner angles are larger than 180 degrees at least on one occasion. It has at least one “caved-in” or indented side. Concave hexagons can have irregular shapes and exhibit more complex geometrical properties compared to convex hexagons. Below is the diagram of an irregular concave Hexagon.Figure-5: Irregular concave hexagon.Relevant FormulaeBelow are the relevant formulas related to a hexagon, specifically a regular hexagon, given that an irregular hexagon does not have simple, universal formulas due to its variable dimensions. For these formulas, we assume a regular hexagon with side lengths ‘s.’PerimeterThe perimeter of a hexagon is simply the sum of the lengths of its sides. For a regular hexagon, all sides are of equal length. So, the perimeter (P) is:P = 6 × sInterior AngleThe total measure of the interior angles of any polygon is given by the formula (n-2) × 180, where ‘n’ is the number of sides. For a hexagon, this gives us (6-2) × 180 = 720 degrees.Each individual interior angle of a regular hexagon (since a regular hexagon’s angles are all equal) is this total divided by the number of sides (6), which gives us 120 degrees.Exterior AngleFor any regular polygon, the measure of each exterior angle is 360/n degrees. In the case of a regular hexagon, this is 360/6 = 60 degrees.AreaThe area of a regular hexagon can be found by dividing it into six equilateral triangles. The area formula is:$$A = \frac{3 \times \sqrt{3}}{2} \times s^2 $$This formula is derived from calculating the area of each equilateral triangle and then multiplying by 6.DiagonalsA hexagon is a six-sided polygon. Every vertex (or corner) of the hexagon is connected to every other vertex by a line segment. Some of these are sides of the hexagon, and the rest are diagonals. A diagonal is a line segment that joins two polygonal vertices that are not contiguous to one another. For a hexagon, each vertex is connected to three other vertices by a diagonal.To find the total number of diagonals, we use the formula:n × (n-3)/2,where n is the number of vertices (or corners) of the polygon.Apothem and RadiusThe apothem of a regular hexagon (a line that connects the hexagon’s centre to any of its side’s midpoints) and the radius (a line from the center to any vertex) are particularly useful in certain calculations.For a regular hexagon with side length ‘s’ Apothem (a) is given by:$$a = \frac{s \times \sqrt{3}}{2} $$Radius (r) = sUsing the Apothem or Radius for AreaAlternatively, the area of a regular hexagon can be found using the apothem (a) or the radius (r) with these formulas:Using the apothem: A = 1/2 × a × P, where P is the perimeter of the hexagon.Using the radius, the area could be calculated as: $$\frac{3}{2} \times r^2 \times \sqrt{3}$$ApplicationsHexagon is a powerful geometric shape with six sides and six angles. It has various applications across different fields due to its unique properties and symmetry. Here, we’ll discuss some of the key applications of hexagons in various domains.Mathematics and GeometryHexagons have long been of interest in mathematics and geometry due to their regularity and tessellating properties. Hexagonal tessellations are highly efficient in covering a plane without any gaps or overlaps. This property makes them useful in fields such as crystallography, where hexagonal lattices are used to represent atomic structures. Hexagons are also integral to the study of hexagonal geometry, which includes topics like hexagonal numbers, hexagonal tilings, and hexagonal grids.Architecture and ConstructionHexagons are aesthetically pleasing shapes that find applications in architectural design. The regular hexagonal shape can be used in the layout of buildings, city planning, and urban design. Honeycomb structures, which consist of hexagonal cells, are known for their strength and efficiency. They are used in various construction applications, such as lightweight composite materials, aerospace components, and structural supports.Manufacturing and EngineeringHexagons play a crucial role in the manufacturing and engineering industries. Hexagonal wrenches and socket sets are widely used for tightening and loosening bolts and screws due to the multiple contact points provided by the six-sided shape. Hex bolts and nuts, known as hex cap screws, are extensively used in machinery, automotive, and construction sectors. The use of hexagons in these fasteners ensures a secure grip and minimizes the risk of stripping or rounding off the corners.Computer Science and Data RepresentationHexadecimal, often referred to as “hex,” is a number system that uses a base of 16. It is frequently employed in digital systems and computer science. Hexadecimal representation allows concise and efficient representation of binary data, particularly in computer memory and programming. Hex codes are used to represent colors in web design, where each color is specified by a six-digit hexadecimal number. Hex grids are also used in computer graphics and game development for representing maps and game worlds.Nature and BiologyHexagons are found abundantly in nature and biology. For example, honeybees construct their hives using hexagonal cells, as it maximizes the use of space while minimizing the amount of wax needed. The efficiency of hexagonal packing can also be observed in the arrangement of cells in plant structures, such as the pattern found in the skin of a pineapple. Additionally, some compounds and molecules, such as benzene, have hexagonal structures.Communication and NetworkingHexagons are employed in the field of wireless communication and networking. Hexagonal cell grids are commonly used in cellular network planning and optimization. The hexagonal layout provides better coverage and capacity compared to other geometric shapes, such as squares or circles. This design allows for a more efficient distribution of base stations and minimizes interference between neighboring cells.In essence, hexagons have a wide range of applications across various fields thanks to their unique geometric properties. From natural formations to human-made structures and systems, their influence and utility are pervasive.ExerciseExample 1Find the sum of the interior angles of a hexagon.SolutionThe formula for the sum of the interior angles of a polygon is (n-2) × 180°, where n is the number of sides. In the case of a hexagon, n = 6.Substituting the value into the formula: (6-2) × 180° = 720°So, the sum of the interior angles of a hexagon is 720°.Example 2A hexagon is regular, which means all of its sides and angles are equal. If one angle measures 120 degrees, is it a regular hexagon?SolutionIn a regular hexagon, each interior angle can be calculated by the formula (n-2) × 180° / n, where n is the number of sides. For a hexagon, n = 6.Substituting n = 6 in the formula gives us: (6-2) × 180° / 6 = 120°So, yes, it is a regular hexagon.Example 3What is the measure of each interior angle of a regular hexagon?SolutionUse the same formula as above: (n-2) × 180° / n. For a hexagon, n = 6.Substitute n = 6 in the formula: (6-2) × 180° / 6 = 120°So, each interior angle of regular hexagon measures 120°.Example 4What is the measure of each exterior angle of a regular hexagon?SolutionThe exterior angle of a regular polygon can be found using the formula 360° / n.For a hexagon, n = 6. Substitute n = 6 into the formula: 360° / 6 = 60°So, each exterior angle of the regular hexagon measures 60°.Example 5For a regular hexagon given in Figure-6, what is its perimeter?Figure-6.SolutionThe perimeter of a regular polygon is calculated by multiplying the length of one side by the number of sides. For a regular hexagon with a side length of 10 units, the perimeter is 6 × 10 = 60 units.So, the perimeter of the hexagon is 60 units.Example 6The area of a regular hexagon is 259.8076 square units. What is the length of each side?SolutionThe area of a regular hexagon can be found using the formula $A = (\frac{3 \times \sqrt{3}}{2} \times s^2 ) $, where s is the length of a side.Rearrange the formula to find s: we get$$s = sqrt{( \frac{A}{\frac{3 \times \sqrt{3}}{2}} )}$$Substitute A = 259.8076 into the formula:$$s = sqrt{( \frac{259.8076}{\frac{3 \times \sqrt{3}}{2}} )}$$s ≈ 10 units.So, each side of the hexagon is approximately 10 units long.Example 7A regular hexagon has a side length of 8 units. What is its area?SolutionUse the area formula for a regular hexagon, $A = \frac{3 \times \sqrt{3}}{2} \times s^2 $.Substitute s = 8 into the formula we get$$A = \frac{3 \times \sqrt{3}}{2} \times 8^2 $$A ≈ 166.2766 square units.So, the area of the hexagon is approximately 166.2766 square units.Example 8A hexagon has 6 vertices. How many diagonals does it have?SolutionThe number of diagonals in a Using the formula n × (n-3)/2, where n is the number of sides, polygons can be computed.For a hexagon, n = 6. Substitute n = 6 into the formula: 6 × (6-3)/2 = 9.So, a hexagon has 9 diagonals. All images were created with GeoGebra.

Posted byWilliam SmithJanuary 10, 2023May 29, 2023Posted inGeometry

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Hexagon - Wikipedia

What is a Hexagon? Definition, Properties, Area, Perimeter, Facts

Hexagons Explained! The Complete Guide to Hexagons — Mashup Math

What is a Hexagon? Definition, Properties, Area, Perimeter, Facts

Hexagons Explained! The Complete Guide to Hexagons — Mashup Math

What is a Hexagon? Definition, Properties, Area, Perimeter, Facts

Hexagon | Definition, Shape, Area, Angles, & Sides | Britannica

HEXAGON中文(简体)翻译：剑桥词典

Hexagon Shape - Sides of Hexagon | Regular Hexagon | Angles of Hexagon

Hexagons Explained! The Complete Guide to Hexagons — Mashup Math

Hexagon

All About Hexagons - Definition, Examples, Formulas - DoodleLearning

Hexagon -- from Wolfram MathWorld

Hexagon - Definition, Geometry, Applications, and Examples

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tokenpocket官网|hexagons

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Hexagon - Wikipedia

What is a Hexagon? Definition, Properties, Area, Perimeter, Facts

Hexagons Explained! The Complete Guide to Hexagons — Mashup Math

What is a Hexagon? Definition, Properties, Area, Perimeter, Facts

Hexagons Explained! The Complete Guide to Hexagons — Mashup Math

What is a Hexagon? Definition, Properties, Area, Perimeter, Facts

Hexagon | Definition, Shape, Area, Angles, & Sides | Britannica

HEXAGON中文(简体)翻译：剑桥词典

Hexagon Shape - Sides of Hexagon | Regular Hexagon | Angles of Hexagon

Hexagons Explained! The Complete Guide to Hexagons — Mashup Math

Hexagon

All About Hexagons - Definition, Examples, Formulas - DoodleLearning

Hexagon -- from Wolfram MathWorld

Hexagon - Definition, Geometry, Applications, and Examples

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