Regular polygon

What is a regular polygon?	A shape with equal sides and equal angles.
Examples	Triangles, squares, and pentagons are all regular polygons.
Fun fact	All regular polygons are also convex shapes.
How many sides?	A regular polygon can have any number of sides — from 3 up to many!

In Euclidean geometry, a regular polygon is a special kind of polygon where every angle is the same size and every side is the same length. This makes regular polygons very neat and symmetrical. They can be either convex, which means all their sides curve outwards, or star, which looks like a star shape.

Regular polygons are important in many areas, such as art, architecture, and nature. For example, honeybees build their hives in shapes made of regular hexagons because this uses space efficiently. As you add more sides to a regular polygon, it starts to look more and more like a circle. This happens whether you keep the total distance around the shape (its perimeter) the same, or you keep the length of each side the same. If the side lengths stay the same and you keep adding sides, the shape stretches out to look like a straight line.

General properties

All regular polygons, whether they are convex or star-shaped, share some special properties. A regular polygon with n sides has rotational symmetry of order n, meaning it looks the same after being rotated by certain angles.

The vertices of a regular polygon all lie on a single circle called the circumscribed circle. Because all sides are equal and all vertices lie on this circle, each regular polygon also has an inscribed circle that touches the midpoint of every side. This makes the polygon both cyclic and tangential.

Symmetry

The symmetry of an n-sided regular polygon is described by a group called the dihedral group D_n. This group includes rotations and reflections. When you rotate the polygon by certain angles, it looks the same. Additionally, the polygon can be reflected across lines that pass through its center. If the number of sides n is even, some of these lines pass through opposite vertices, and others pass through the midpoints of opposite sides. If n is odd, each line passes through one vertex and the midpoint of the opposite side.

Regular convex polygons

All regular simple polygons are convex, meaning they don’t intersect themselves. Regular polygons with the same number of sides are also similar, meaning they have the same shape but can be different sizes.

For a regular polygon with n sides, each interior angle measures (\frac{(n-2) \times 180^\circ}{n}). As the number of sides increases, the shape gets closer to a circle. For example, a regular polygon with 10,000 sides has an interior angle of about 179.96°, very close to a full 180°.

Examples of dissections for selected even-sided regular polygons

Sides	6	8	10	12	14	16
Rhombs	3	6	10	15	21	28


Sides	18	20	24	30	40	50
Rhombs	36	45	66	105	190	300

Number of sides	Area when side s = 1		Area when circumradius R = 1			Area when apothem a = 1
Number of sides	Exact	Approximation	Exact	Approximation	Relative to circumcircle area	Exact	Approximation	Relative to incircle area
n	n 4 cot ⁡ ( π n ) {\displaystyle \scriptstyle {\tfrac {n}{4}}\cot \left({\tfrac {\pi }{n}}\right)}		n 2 sin ⁡ ( 2 π n ) {\displaystyle \scriptstyle {\tfrac {n}{2}}\sin \left({\tfrac {2\pi }{n}}\right)}		n 2 π sin ⁡ ( 2 π n ) {\displaystyle \scriptstyle {\tfrac {n}{2\pi }}\sin \left({\tfrac {2\pi }{n}}\right)}	n tan ⁡ ( π n ) {\displaystyle \scriptstyle n\tan \left({\tfrac {\pi }{n}}\right)}		n π tan ⁡ ( π n ) {\displaystyle \scriptstyle {\tfrac {n}{\pi }}\tan \left({\tfrac {\pi }{n}}\right)}
3	⁠ 3 4 {\displaystyle \scriptstyle {\tfrac {\sqrt {3}}{4}}} ⁠	0.433012702	⁠ 3 3 4 {\displaystyle \scriptstyle {\tfrac {3{\sqrt {3}}}{4}}} ⁠	1.299038105	0.4134966714	⁠ 3 3 {\displaystyle \scriptstyle 3{\sqrt {3}}} ⁠	5.196152424	1.653986686
4	1	1.000000000	2	2.000000000	0.6366197722	4	4.000000000	1.273239544
5	⁠ 1 4 25 + 10 5 {\displaystyle \scriptstyle {\tfrac {1}{4}}{\sqrt {25+10{\sqrt {5}}}}} ⁠	1.720477401	⁠ 5 4 1 2 ( 5 + 5 ) {\displaystyle \scriptstyle {\tfrac {5}{4}}{\sqrt {{\tfrac {1}{2}}\left(5+{\sqrt {5}}\right)}}} ⁠	2.377641291	0.7568267288	⁠ 5 5 − 2 5 {\displaystyle \scriptstyle 5{\sqrt {5-2{\sqrt {5}}}}} ⁠	3.632712640	1.156328347
6	⁠ 3 3 2 {\displaystyle \scriptstyle {\tfrac {3{\sqrt {3}}}{2}}} ⁠	2.598076211	⁠ 3 3 2 {\displaystyle \scriptstyle {\tfrac {3{\sqrt {3}}}{2}}} ⁠	2.598076211	0.8269933428	⁠ 2 3 {\displaystyle \scriptstyle 2{\sqrt {3}}} ⁠	3.464101616	1.102657791
7		3.633912444		2.736410189	0.8710264157		3.371022333	1.073029735
8	⁠ 2 + 2 2 {\displaystyle \scriptstyle 2+2{\sqrt {2}}} ⁠	4.828427125	⁠ 2 2 {\displaystyle \scriptstyle 2{\sqrt {2}}} ⁠	2.828427125	0.9003163160	⁠ 8 ( 2 − 1 ) {\displaystyle \scriptstyle 8\left({\sqrt {2}}-1\right)} ⁠	3.313708500	1.054786175
9		6.181824194		2.892544244	0.9207254290		3.275732109	1.042697914
10	⁠ 5 2 5 + 2 5 {\displaystyle \scriptstyle {\tfrac {5}{2}}{\sqrt {5+2{\sqrt {5}}}}} ⁠	7.694208843	⁠ 5 2 1 2 ( 5 − 5 ) {\displaystyle \scriptstyle {\tfrac {5}{2}}{\sqrt {{\tfrac {1}{2}}\left(5-{\sqrt {5}}\right)}}} ⁠	2.938926262	0.9354892840	⁠ 2 25 − 10 5 {\displaystyle \scriptstyle 2{\sqrt {25-10{\sqrt {5}}}}} ⁠	3.249196963	1.034251515
11		9.365639907		2.973524496	0.9465022440		3.229891423	1.028106371
12	⁠ 6 + 3 3 {\displaystyle \scriptstyle 6+3{\sqrt {3}}} ⁠	11.19615242	3	3.000000000	0.9549296586	⁠ 12 ( 2 − 3 ) {\displaystyle \scriptstyle 12\left(2-{\sqrt {3}}\right)} ⁠	3.215390309	1.023490523
13		13.18576833		3.020700617	0.9615188694		3.204212220	1.019932427
14		15.33450194		3.037186175	0.9667663859		3.195408642	1.017130161
15	⁠ 15 8 ( 15 + 3 + 2 ( 5 + 5 ) ) {\displaystyle \scriptstyle {\tfrac {15}{8}}\left({\sqrt {15}}+{\sqrt {3}}+{\sqrt {2\left(5+{\sqrt {5}}\right)}}\right)} ⁠	17.64236291	⁠ 15 16 ( 15 + 3 − 10 − 2 5 ) {\displaystyle \scriptstyle {\tfrac {15}{16}}\left({\sqrt {15}}+{\sqrt {3}}-{\sqrt {10-2{\sqrt {5}}}}\right)} ⁠	3.050524822	0.9710122088	⁠ 15 2 ( 3 3 − 15 − 2 ( 25 − 11 5 ) ) {\displaystyle \scriptstyle {\tfrac {15}{2}}\left(3{\sqrt {3}}-{\sqrt {15}}-{\sqrt {2\left(25-11{\sqrt {5}}\right)}}\right)} ⁠	3.188348426	1.014882824
16	⁠ 4 ( 1 + 2 + 2 ( 2 + 2 ) ) {\displaystyle \scriptstyle 4\left(1+{\sqrt {2}}+{\sqrt {2\left(2+{\sqrt {2}}\right)}}\right)} ⁠	20.10935797	⁠ 4 2 − 2 {\displaystyle \scriptstyle 4{\sqrt {2-{\sqrt {2}}}}} ⁠	3.061467460	0.9744953584	⁠ 16 ( 1 + 2 ) ( 2 ( 2 − 2 ) − 1 ) {\displaystyle \scriptstyle 16\left(1+{\sqrt {2}}\right)\left({\sqrt {2\left(2-{\sqrt {2}}\right)}}-1\right)} ⁠	3.182597878	1.013052368
17		22.73549190		3.070554163	0.9773877456		3.177850752	1.011541311
18		25.52076819		3.078181290	0.9798155361		3.173885653	1.010279181
19		28.46518943		3.084644958	0.9818729854		3.170539238	1.009213984
20	⁠ 5 ( 1 + 5 + 5 + 2 5 ) {\displaystyle \scriptstyle 5\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}\right)} ⁠	31.56875757	⁠ 5 2 ( 5 − 1 ) {\displaystyle \scriptstyle {\tfrac {5}{2}}\left({\sqrt {5}}-1\right)} ⁠	3.090169944	0.9836316430	⁠ 20 ( 1 + 5 − 5 + 2 5 ) {\displaystyle \scriptstyle 20\left(1+{\sqrt {5}}-{\sqrt {5+2{\sqrt {5}}}}\right)} ⁠	3.167688806	1.008306663
100		795.5128988		3.139525977	0.9993421565		3.142626605	1.000329117
1000		79577.20975		3.141571983	0.9999934200		3.141602989	1.000003290
10⁴		7957746.893		3.141592448	0.9999999345		3.141592757	1.000000033
10⁶		79577471545		3.141592654	1.000000000		3.141592654	1.000000000

Constructible polygon

Main article: Constructible polygon

Some regular polygons can be drawn using just a compass and straightedge, while others cannot. Ancient Greek mathematicians could make polygons with 3, 4, or 5 sides, and they learned how to double the number of sides from any regular polygon they already had. This made them wonder which regular polygons could be constructed this way.

In 1796, Carl Friedrich Gauss showed that a regular 17-gon could be constructed. He created a rule to determine which regular polygons can be made: a regular n-gon can be constructed if n is made by multiplying a power of 2 by any number of special primes called Fermat primes. Later, another mathematician named Pierre Wantzel proved that this rule also explained which polygons could not be constructed.

Regular skew polygons

A regular skew polygon in 3-space is a special shape that moves in a zigzag way between two parallel flat surfaces. It is formed by the edges of a shape called a uniform antiprism, and all its sides and angles are the same.

More generally, regular skew polygons can exist in any number of dimensions. Examples include the Petrie polygons, which are paths along edges that split a regular polytope into two parts. When these polygons become very large, they turn into shapes called skew apeirogons.

The cube contains a skew regular hexagon, seen as 6 red edges zig-zagging between two planes perpendicular to the cube's diagonal axis.

The zig-zagging side edges of a n-antiprism represent a regular skew 2n-gon, as shown in this 17-gonal antiprism.

Regular star polygons

A regular star polygon is a special type of non-convex regular polygon. The most well-known example is the pentagram, which looks like a five-pointed star and shares the same vertices as a pentagon, but connects every second vertex.

For an n-sided star polygon, we use a special notation called the Schläfli symbol, written as {n/m}, where m shows how many steps we take to connect the vertices. If m is 2, we connect every second point; if m is 3, we connect every third point, and so on. The lines of the star polygon wrap around the center m times.

Some common regular star polygons with up to 12 sides include:

Pentagram – {5/2}
Heptagram – {7/2} and {7/3}
Octagram – {8/3}
Enneagram – {9/2} and {9/4}
Decagram – {10/3}
Hendecagram – {11/2}, {11/3}, {11/4} and {11/5}
Dodecagram – {12/5}

For these shapes to be true stars and not collapse into simpler shapes, n and m must have no common factors other than 1, meaning they are coprime.

Two interpretations of {6/2}
Grünbaum {6/2} or 2{3}	Coxeter 2{3} or {6}[2{3}]{6}

Doubly-wound hexagon	Hexagram as a compound of two triangles

Duality of regular polygons

All regular polygons are self-dual, meaning they look the same even when you flip them. For polygons with an odd number of sides, they look exactly the same when flipped. Regular star figures, which are made up of regular polygons, are also self-dual.

Regular polygons as faces of polyhedra

A uniform polyhedron is a special 3D shape that uses regular polygons as its faces. This means that all the shapes on its surface are the same size and shape, and you can move one part of the shape to match any other part by turning or flipping it.

There are also other interesting 3D shapes that use regular polygons. Some of these, called quasiregular polyhedra, have two different kinds of regular polygon faces that take turns around each corner. When a 3D shape has only one kind of regular polygon face, it is known as a regular polyhedron. Other shapes with regular polygon faces that are not uniform are called Johnson solids. If a polyhedron has only regular triangles as faces, it is known as a deltahedron.