The formula we use to find the sum of the interior angles of any polygon comes from the
following idea:
Notice that a pentagon has 5 sides, and that you can form 3 triangles by
connecting the vertices. That's 2 less than the number of sides. It's the
same principle for all polygons.
If we represent the number of sides of a polygon as n, then the number of triangles
you can form is (n-2). Since each triangle contains 180 , that gives us the formula:
0
Sum of Interior Angles = 180(n-2)
There are two types of problems for which we will be using this formula:
1. Questions that ask you to find the number of degrees in the sum of the interior
angles of a polygon.
2. Questions that ask you to find the number of degrees in one angle of a regular
polygon.
Hint: When working with the angle formulas for polygons, be sure to read each question carefully for
clues as to which formula you will need to use to solve the problem. Look for the words that describe each
formula, such as the words sum, interior, and angles.
Example 1: Find the number of degrees in the sum of the interior angles of an
octagon.
Example 2: What is the measure of each interior angle of a regular octagon?
A regular polygon's sides are all of the same length and its angles are the same size.
Regular polygon
shape
Name &
# of sides
(n)
# of triangles
( n  2)
triangle
n3
square
n  __
Pentagon
n  __
Hexagon
n  __
Heptagon
n  __
Octagon
n  __
Nonagon
n  __
Decagon
n  __
3 2 1
Sum of interior
angles
180(n  2)
180(n  2)
180(3  2)
180(1)
180o
180(n  2)
__  2  __
180(4  2)
180(___)
180( n  2)
180(__  2)
180(___)
Each interior angle
measure
180( n  2)
n
180(3  2)
=
3
180(1)
=
3
60o
6.3 Skills Practice, #1-12 (Find the sum of the measures of the interior angles of each polygon)
use:
Sum of Interior Angles = 180(n-2)
1) 13-gon
2) 17-gon
3) 18-gon
4) 24-gon
5) 32-gon
6) 35-gon
7) 21-gon
8) 29-gon
9) 54-gon
10) 64-gon
11) 81-gon
12) 150-gon
6.3 Skills Practice, #13-24 (Find measure of one interior angles of the given regular polygon-
13) heptagon (7-sided)
180( n  2)
n
14) 26-gon
15) decagon (10-sided)
16) 23-gon
17) 37-gon
18) 51-gon
19) 48-gon
20) 85-gon
21) 72-gon
22) 49-gon
23) 66-gon
24) 500-gon
round to the nearest tenth if necessary) use: