Area of Regular Polygons
A regular polygon, remember, is a
polygon whose sides
and interior angles are all
congruent. To understand the formula for the
area of such a polygon, some new vocabulary is necessary.
The center of a regular polygon is the
point from which all the
vertices are
equidistant.
The radius of a regular polygon is a
segment with one endpoint at the center and
the other endpoint at one of the vertices. Thus, there are n radii in an
n-sided regular polygon. The center and radius of a regular polygon are the same
as the center and radius of a circle
circumscribed about that regular polygon.
An apothem of a regular polygon is a segment with one endpoint at the center
and the other endpoint at the midpoint of one
of the sides. The apothem of a regular polygon is the perpendicular
bisector of whichever side on which it has its
endpoint. A central angle of a regular polygon is an
angle whose
vertex is the center and whose
rays, or sides, contain the endpoints of a
side of the regular polygon. Thus, an n-sided regular polygon has n apothems
and n central angles, each of whose measure is 360/n
degrees. Every apothem is the angle
bisector of the central angle that contains the
side to which the apothem extends. Below are pictured these characteristics of
a regular polygon.
Figure 6.1: A regular polygon with a center (C), radius (r), apothem (a), and
central angle
Once you have mastered these new definitions, the formula for the area of a
regular polygon is an easy one. The area of a regular polygon is one-half the
product of its apothem and its perimeter. Often the formula is written like
this: Area=1/2(ap), where a denotes the length of an apothem, and p denotes the
perimeter.
When an n-sided polygon is split up into n
triangles, its area is equal to the sum of the
areas of the triangles. Can you see how 1/2(ap) is equal to the sum of the
areas of the triangles that make up a regular polygon? The apothem is equal to
the altitude, and the perimeter is equal to the sum of the
bases. So
1/2(ap) is only a slightly simpler way to express the sum of the areas of the n
triangles that make up an n-sided regular polygon.
Figure 6.2: Two n-sided polygons divided into n triangles
