Regular polygons
A regular polygon is a polygon which is equiangular (all angles are congruent) and equilateral (all sides have the same length). Regular polygons may be convex or star.
These properties apply to both convex and a star regular polygons.
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.
Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle.
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.
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.
All regular simple polygons (a simple polygon is one which does not intersect itself anywhere) are convex. Those having the same number of sides are also similar.
An n-sided convex regular polygon is denoted by its Schläfli symbol {n}.
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, etc.
For a regular convex n-gon, each interior angle has a measure of:
and each exterior angle (supplementary to the interior angle) has a measure of 
degrees, with the sum of the exterior angles equal to 360 degrees or 2Ï€ radians or one full turn.
For n > 2 the number of diagonals is 
, i.e., 0, 2, 5, 9, ... They divide the polygon into 1, 4, 11, 24, ... pieces.
The area A of a convex regular n-sided polygon is:
in degrees
or in radians
where t is the length of a side.
If the radius, or length of the segment joining the center to the vertex is known, the area is:
in degrees
or in radians
where r is the radius
Also, the area is half the perimeter multiplied by the length of the apothem, a, (the line drawn from the centre of the polygon perpendicular to a side). That is A = a.n.t/2, as the length of the perimeter is n.t, or more simply 1/2 p.a.
For sides t=1 this gives:
in degrees
or in radians (n not equal to 2)
with the following values:
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).
A non-convex regular polygon is a regular star polygon. The most common example is the pentagram, which has the same vertices as a pentagon, but connects alternating vertices.
For an n-sided star polygon, the Schläfli symbol is modified to indicate the 'starriness' m of the polygon, as {n/m}. If m is 2, for example, then every second point is joined. If m is 3, then every third point is joined. The boundary of the polygon winds around the centre m times, and m is sometimes called the density of the polygon.
Examples:
m and n must be co-prime, or the figure will degenerate. Depending on the precise derivation of the Schläfli symbol, opinions differ as to the nature of the degenerate figure. For example {6/2} may be treated in either of two ways:
All regular polygons are self-dual, with equal number of vertices and edges.
In addition the regular star polygons and regular star figures (compounds), being composed of regular polygons, are also self-dual.
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 (just as there is for a regular polygon).
The remaining convex polyhedra with regular faces are known as the Johnson solids.
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Last updated on Monday September 24, 2007 at 05:23:47 PDT (GMT -0700)
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