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Polygons—Formulas
Geometry CP2 (Holt 6-1) K.Santos
Polygon Interior Angle Sum
The sum of the measures of the angles of an n-gon
is (n-2)180°.
Example: 16-gon
(n-2)180° with n=16
(16-2)180
(14)180
2520°
Why does this formula work?
1 triangle
180°
2 triangles
360°
3 triangles
540°
1(180)
2(180)
3(180)
(3-2)180
(4-2)180
(5-2)180
first number in the formula is number of sides
(n-2)180°
Example—Finding a missing angle
Find the missing angle measures.
First you have to find out what
the interior angle sum is.
(n-2)180°
(5-2)180
3(180)
540°
Then write an equation to solve for the angles:
121+112+78+x+x+3= 540
314 +2x = 540
2x = 226
x = 113
So the angles measure: 113° and 116°
(x and x + 3)
121°
112°
78°
x
x+3
Polygon—Exterior Angle Sum
The sum of the measures of the exterior angles of a
polygon, one at each vertex, is 360°.
m<1+m<2+m<3+m<4+m<5 =360°
1
5
2
3
4
Polygon—Exterior Angle Sum
For any polygon the exterior angle
sum is 360°.
So, if it is a 50-gon then it’s exterior angle sum is
360°.
So, if it is a dodecagon then it’s exterior angle sum
is 360°.
Regular Polygon—One/each interior
angle
To find one (or each) interior angle in a regular
polygon with n, number of sides:
𝑡𝑜𝑡𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒 𝑠𝑢𝑚
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑖𝑑𝑒𝑠
(n−2)180°
𝑛
Example—Each interior angle in a
regular polygon
Find the measure of each interior angle of a
regular 18-gon.
(n−2)180°
𝑛
(18−2)180°
18
(16)180°
2880
18
18
each angle measures: 160°
Regular Polygon—One/each exterior
angle
To find one (or each) exterior angle in a regular
polygon with n, number of sides:
𝑡𝑜𝑡𝑎𝑙 𝑒𝑥𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒 𝑠𝑢𝑚
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑖𝑑𝑒𝑠
360°
𝑛
Example—Each exterior angle in a
regular polygon
Find the measures of each exterior angle in a
regular 24-gon.
360°
𝑛
360°
24
each angle measures: 15°
Regular Polygon—Interior and Exterior
Angle Relationships
Find each interior and exterior angle in a regular 40-gon.
Interior Angle
(n−2)180°
𝑛
(40−2)180°
40
(38)180°
40
6840°
40
Exterior Angle
360°
𝑛
360°
40
9° exterior angle
171° interior angle
Notice that the interior angle and exterior angle are supplementary
(add to 180°).
Exterior angle
interior angle