Regular polygon
![](http://upload.wikimedia.org/wikipedia/commons/thumb/0/04/Pentagon.svg/220px-Pentagon.svg.png)
A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length).
All regular polygons with the same number of sides are similar.
- Regular digon: degenerate, a "double line segment"
- Equilateral triangle (3 side)
- Square (4 sides)
- Regular pentagon (5 sides)
- Regular hexagon (6 sides)
- Regular heptagon (7 sides)
- Regular octagon (8 sides)
- Regular decagon (10 sides)
- Regular dodecagon (12 sides)
In certain contexts all the polygons considered will be regular. In such circumstances it is customary to drop the prefix regular. For instance all the faces of uniform polyhedra must be regular and the faces will be described simply as triangle, square, pentagon.
Properties
Each angle of a regular n-gon has a measure of (or equally of ) degrees.
Alternately, the internal angle(s) of a regular n-gon is radians ( or turns).
All vertices of a regular polygon lie on a common circle, i.e., they are concyclic points, i.e., every regular polygon has a circumscribed circle.
A regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. See constructible polygon.
For the number of diagonals is , i.e., 0, 2, 5, 9, ... They divide the polygon into 1, 4, 11, 24, ... pieces.
Area
![](http://upload.wikimedia.org/wikipedia/commons/thumb/7/73/Apothem_of_hexagon.svg/220px-Apothem_of_hexagon.svg.png)
The area of a regular n-sided polygon is
where t is the length of a side. Also, the area is half the perimeter multiplied by the length of the apothem (the line drawn from the centre of the polygon perpendicular to a side).
For t=1 this gives
with the following values:
Sides | Name | Exact area | Approximate area |
---|---|---|---|
3 | equilateral triangle | 0.433 | |
4 | square | 1 | 1.000 |
5 | regular-pentagon | 1.720 | |
6 | regular-hexagon | 2.598 | |
7 | regular-heptagon | 3.634 | |
8 | regular-octagon | 4.828 | |
9 | regular-enneagon | 6.182 | |
10 | regular-decagon | 7.694 | |
11 | regular-hendecagon | 9.366 | |
12 | regular-dodecagon | 11.196 | |
13 | regular-triskaidecagon | 13.186 | |
14 | regular-tetradecagon | 15.335 | |
15 | regular-pentadecagon | 17.642 | |
16 | regular-hexadecagon | 20.109 | |
17 | regular-heptadecagon | 22.735 | |
18 | regular-octadecagon | 25.521 | |
19 | regular-enneadecagon | 28.465 | |
20 | regular-icosagon | 31.569 | |
100 | regular-hectagon | 795.513 | |
1000 | regular-chiliagon | 79577.210 | |
10000 | regular-myriagon | 7957746.893 |
The amounts that the areas are less than those of circles with the same perimeter, are (rounded) equal to 0.26, for n<8 a little more (the amounts decrease with increasing n to the limit π/12).
Symmetry
The symmetry group of an n-sided regular polygon is dihedral group Dn (of order 2n): D2, D3, D4,... It consists of the rotations in Cn (there is rotational symmetry of order n), together with reflection symmetry in n axes that pass through the center. If n is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If n is odd then all axes pass through a vertex and the midpoint of the opposite side.
Nonconvex regular polygons
![](http://upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Pentagram_green.svg/150px-Pentagram_green.svg.png)
An extended category of regular polygons includes the star polygons, for example a pentagram, which has the same vertices as a pentagon, but connects alternating vertices.
Examples:
- Pentagram - {5/2}
- Heptagram - {7/2}, {7/3}
- Octagram - {8/3}
- Enneagram - {9/2}, {9/4}
- Decagram - {10/3}
Polyhedra
A uniform polyhedron is a polyhedron with regular polygons as faces such that for every two vertices there is an isometry mapping one into the other.
See also
External links
- Weisstein, Eric W. "Regular polygon". MathWorld.
- Regular Polygon description With interactive animation
- Incircle of a Regular Polygon With interactive animation
- Area of a Regular Polygon Three different formulae, with interactive animation