notes
Geometry
Polygon Angle Formulas
INTERIOR ANGLES OF POLYGONS
Sum of the INTERIOR Angle Measures of Number of sides of ANY Polygon when
given the sum of the INTERIOR angles:
ANY Polygon:
S = 180(n − 2)
S = sum of all interior angles
To get the formula on the
right, just solve the
equation on the left for n.
n = number of sides
S
n=
+2
180
S = sum of all interior angles
n = number of sides
If in words or a diagram you are given all of the angle measures of a figure and you
are given (or can see) how many sides the figure has and asked to find a missing
measure:
This figure has
four sides so the
1. Find the sum of the interior angles
sum of the angles
2. Add up the given angles and set it equal to
must be 360°
this sum
79° + 110° + 55° + x = 360°
3. At this point you may need to solve for a
244° + x = 360°
missing angle or variable to find the solution.
x = 116°
EXTERIOR ANGLES OF POLYGONS
The Sum of the exterior angles of a
polygon is 360° regardless of how
many sides it has.
Find the value of x in
the polygon.
1. Add all the exterior angles together.
(Make sure if there are 6 interior angles, that there are 6
exterior angles)
2. Set the sum you found in step 1 equal to 360 and
solve.
46 + 3𝑥 + 62 + 4𝑥 + 47 + 93 = 360
248 + 7𝑥 = 360
7𝑥 = 112
𝑥 = 16
INTERIOR ANGLES OF REGULAR POLYGONS
** In Regular Polygons all sides are the same length, and all Interior angles have
the same measure, and all Exterior angles have the same measure**
The measure of a single interior angle in a regular polygon can be found by dividing
the sum of the interior angle measures, S, by the number of sides, n.
Measure of a single
interior angle in a
regular polygon
=
Sum of all interior angles
number of sides
180(n − 2)
=
𝑛
Measure of a single
interior angle in a
regular polygon
Example 1: What is the measure of one interior angle in a regular 24-gon.
Using the middle formula:
180(24 − 2) 180(22) 3960
=
= 165°
=
24
24
24
Example 2: The measure of an interior angle of a regular polygon is 144°. Find
the number of sides of the polygon.
Measure of a single
180(n − 2) So… for this problem 144 = 180(n − 2)
interior angle in a =
n
regular polygon
𝑛
1. Multiply both sides by n:
To solve:
144 =
180(n − 2)
n
𝑛 ∙ 144 =
180(n − 2)
∙X
𝑛
nX
2. Distribute the 180:
3. Solve for n:
144n = 180n − 360 −36𝑛 = −360
n = 10
If a single interior angle measure of a regular polygon is 144°, the polygon will have 10 sides.
EXTERIOR ANGLES OF A REGULAR POLYGON
Example 1: What is each angle measure of a regular 18-gon?
Each Exterior
Angle Measure
of a REGULAR =
Polygon
360
= 20°
18
360
n
Each exterior angle measure in a REGULAR 18-gon is 20°
Example 2: If the exterior angle measure of a regular polygon is 12°, how
many sides does the polygon have?
Number
360
=
of sides
One exterior angle measure
360
= 30 sides
12
If an exterior angle measure in a REGULAR
Polygon is 12°, the Polygon must have 30 sides.