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 $$\frac{a}{2}$$.