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 jointing 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, divides it into as many equal isosceles triangles as there are sides in it.

The angle of a regular polygon of sides .
Detail of Sides of Polygon:
There is no theoretical limit to the number of sides of a polygon, but only those with twelve or less are commonly met with. The names of polygons which are mostly in use 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:
Perimeter of a regular polygon = perimeter of square
A regular polygon can be divided into congruent triangles having common vertex at the center of the polygon. The number of these congruent triangles is the same as that of its sides.
Area of one such triangle sides of polygon length of perpendicular from the center to any side of polygon.
Area of one such triangle, being length of perpendicular
Area of Polygon = Sum of areas of all such triangles
Area of Polygon Perimeter of polygon
Area of Polygon  (1)
Now, Area of Polygon = Area of Square (given)  (2)
From (1) and (2), we have
Hence, the length of perpendicular is .