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.

\therefore the area of one such triangle = \frac{1}{2}ah, h being the length of the perpendicular

\therefore Area of polygon = Sum of areas of all such triangles

\therefore Area of polygon  = \frac{1}{2} \times Perimeter of polygon  \times {\text{ }}h

\therefore Area of polygon  = \frac{1}{2} \times 4a \times h = 2ah --- (1)

Now, the area of the polygon = Area of square = {a^2} (given) --- (2)

\therefore from (1) and (2), we have
\therefore 2ah = {a^2} \Rightarrow h = \frac{a}{2}

 

Hence, the length of the perpendicular is \frac{a}{2}.