A polygon is said to be regular if and only if the following conditions are satisfied:
(1) All the sides are of equal length.
(2) All the interior angles are equal.
The Unique Shape Theorem states that all regular polygons with the same number of sides are similar to each other. And the immediate result of this theorem is that there is one unique shape for them.
Proof:
Condition (2) guarantees that all regular polygons with the same number of sides are similar.
Center
The center of a regular polygon is defined as the point inside the polygon which is equidistant to all the vertexes of it.
Radius
The radii of a regular polygon are defined as the lines joining the vertexes and the center.
The Separable Theorem states that all regular n-sided polygons are separable into n congruent (identical) isosceles triangles.
Proof:
Since there is only one unique shape for a set of regular polygons with the same number of sides(The Unique Shape Theorem), if we are able to describe an algorithm to construct an n-sided regular polygon by using congruent isosceles triangles only, we can prove the theorem.
To begin with, let's consider a Cartesian xy-plane. Say, we now want to construct an n-sided regular polygon. We first radiate n equal length lines from the origin. Each of the lines are separated by an angle of 2×pi/n. Then we join all the vertexes together. As you will see, all the triangle formed are isosceles and congruent(SAS). This completes the proof.
Area of an n-sided regular polygon = 1/2 × n×r²×sin(2×pi/n)
Proof:
From The Separable Theorem, we know that the an n-sided regular polygon can be separated into n congruent isosceles triangles. This follows that
And by the simple triangle area formula, 1/2 × a × b × sin(theta)
In this case, a = b = r (radius), theta = 2×pi/n
Therefore, Area of an isosceles triangle = 1/2 × r²×sin(2×pi/n)
And hence,
The magnitude of an interior angle of an n-sided regular polygon
= pi - 2×pi/n
The magnitude of an exterior angle of an n-sided regular polygon
= pi + 2×pi/n
Proof:
It can be observed that the magnitude of an interior angle of an n-sided regular polgygon is double the magnitude of its respective isosceles triangle's base angle. Therefore, we have
When the number of sides of a regular polygon is infinity, this regular polygon is a circle.
Proof:
The following is a Java applet showing how a regular polygon evolving into a circle.
Mathematically, by the limit theorems, we have
lim magnitude of an interior angle of an n-sided regular polygon
n->inf
= lim (pi + 2×pi/n)
n->inf
= pi
Similarly,
lim magnitude of an exterior angle of an n-sided regular polygon
n->inf
= pi
The above calculations implies the existence of tangent which touches a point (or a vertex if we think a circle in terms of a infinite-sided regular polygon) having the interior angle equals to the exterior angle equals pi. Further evidence is the fact that the area of the infinite-sided regular polygon is in fact identical to the area of a circle.
lim 1/2 × n×r²sin(2×pi/n)
n->inf
= lim (1/2 ×r²×2×pi/n)×(sin(2×pi/n)/(2×pi/n))
n->inf
(Recall lim sin(theta)/theta = 1)
theta->0
= pi×r²