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 sides .
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


Pentagon


Hexagon


Heptagon


Octagon


Nonagon


Decagon






gon

Example:
The perimeter and area of a regular polygon are respectively equal to those of a square of sides . 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
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.
the area of one such triangle sides of polygon length of perpendicular from the center to any side of polygon.
the area of one such triangle, being the length of the perpendicular
Area of polygon = Sum of areas of all such triangles
Area of polygon Perimeter of polygon
Area of polygon  (1)
Now, the area of the polygon = Area of square (given)  (2)
from (1) and (2), we have
Hence, the length of the perpendicular is .