Angles of Polygons
Example 1 Interior Angles of Regular Polygons
ARCHITECTURE The Pentagon in Washington, D.C. is shaped like a regular pentagon. Find the
sum of the measures of the interior angles of the largest pentagon-shaped section of the Pentagon
building.
Since the Pentagon is a convex polygon, we can
use the Interior Angle Sum Theorem.
S = 180(n - 2) Interior Angle Sum Theorem
= 180(5 - 2) n = 5
= 180(3) or 540 Simplify.
The sum of the measures of the interior angles is 540.
Example 2 Sides of a Polygon
The measure of an interior angle of a regular polygon is 135. Find the number of sides of the
polygon.
Use the Interior Angle Sum Theorem to write an equation to solve for n, the number of sides.
S = 180(n - 2) Interior Angle Sum Theorem
(135)n = 180(n - 2) S = 135n
135n = 180n - 360 Distributive Property
360 = 45n Subtract 135n and add 360 to each side.
8 = n Divide each side by 45.
The polygon has 8 sides.
Example 3 Interior Angles of Nonregular Polygons
ALGEBRA Find the measure of each interior angle.
Since n = 4, the sum of the measures of the interior angles
is 180(4 – 2) or 360. Write an equation to express the sum
of the measures of the interior angles of the polygon.
360 = m R + m S + m T + m U Sum of measures of angles
360 = x + 2x + 3x + 4x Substitution
360 = 10x Combine like terms.
36 = x Divide each side by 10.
Use the value of x to find the measure of each angle.
m R = 36, m S = 2 ∙ 36 or 72, m T = 3 ∙ 36 or 108, m U = 4 ∙ 36 or 144.
Example 4 Exterior Angles
Find the measures of an exterior angle and an interior
angle of convex regular nonagon ABCDEFGHI.
At each vertex, extend a side to form one exterior angle. The sum
of the measures of the exterior angles is 360. A convex regular
nonagon has 9 congruent exterior angles.
9n = 360 n = measure of each exterior angle
n = 40 Divide each side by 9.
