Properties of a Regular Polygon

These are the properties of a regular polygon:

The sides and interior angles of a regular polygon are all equal.
The bisectors of the interior angles of a regular polygon meet at its center.
The perpendiculars drawn from the center of a regular polygon to its sides are all equal.
The lines joining the center of a regular polygon to its vertices are all equal.
The center of a regular polygon is the center of both the inscribed and circumscribed circles.
Straight lines drawn from the center to the vertices of a regular polygon divide it into as many equal isosceles triangles as there are sides in it.
The angle of a regular polygon of $n$ sides $= \left( {\frac{{2n - 4}}{n}} \right) \times {90^ \circ }$ .

Detail of the Sides of a Polygon

There is no theoretical limit to the number of sides of a polygon, but only those with twelve or less are commonly seen. The names of polygons which are used most often are as follows:

Number of sides	Polygon name
$5$	Pentagon
$6$	Hexagon
$7$	Heptagon
$8$	Octagon
$9$	Nonagon
$10$	Decagon
$\cdots$	$\cdots$
$\cdots$	$\cdots$
$n$	$n -$ gon

Example:

The perimeter and area of a regular polygon are respectively equal to those of a square of sides $a$ . Find the length of the perpendicular from the center of a regular polygon to any of its sides.

Solution:

The perimeter of a regular polygon = perimeter of square $= 4a$

A regular polygon can be divided into congruent triangles having a common vertex at the center of the polygon. The number of these congruent triangles is the same as that of its sides.

$\therefore$ the area of one such triangle $= \frac{1}{2} \times$ sides of polygon $\times$ length of perpendicular from the center to any side of polygon.