Idiot\'s Guides Basic Math and Pre-Algebra

Chapter 14: Quadrilaterals 195

A regular polygon has a center point, and the radius, or distance from the center to a vertex, is
the same no matter what vertex you go to. If you draw all the radii, you break the polygon into
congruent triangles, one for each side of the polygon. If you can find the area of one of those
triangles, you can multiply by the number of sides to find the area of the polygon.

Each of the triangles is isosceles, because all the radii are equal. The altitude from the center
vertex to a side of the polygon is called the apothem, and it’s a super segment that is a median and
angle bisector as well as an altitude. That means that if you know the length of a side and the
radius, you can find the length of the apothem. Or if you know a side and the apothem, you can
find the radius. Or if you know the radius and the apothem, you can find the side. How? The
Pythagorean theorem, a^2 + b^2 = c^2. When you draw the radii and an apothem, you create a right
triangle whose legs are the apothem and half a side and whose diagonal is the radius. If you know
any two of those, you can find the third.

DEFINITION

The apothem of a regular polygon is a line segment from the center of the polygon

perpendicular to a side.

Suppose you have a regular pentagon with an apothem equal to 4 cm and a radius of 5 cm.
You could use the Pythagorean theorem to find out that half the side is 3 cm, so the whole side
is 6 cm. Once you have the measures of the apothem and a side, you can find the area of one

triangle. A = (
1
2 × 6) × 4 = 12 square centimeters. There are five of those triangles, so the area of
the pentagon is 5 v 12 = 60 square centimeters.

The area of a regular polygon with n sides of length s and apothem a is given by the formula

A nas^1
2

 , but since nas ans^1

2

1

2

 and ns is the perimeter of the regular polygon, the formula

is often given as A =

1

2 aP where P is the perimeter.