1. Formulas Involving Polygons Chapter 7 Section 3
By: Alex Pipcho

2. Polygon Names 8 sides – Octagon 3 sides – Triangle 9 sides- Nonagon
10 sides – Decagon
12 sides – Dodecagon
15 sides –Pentadecagon
n sides – n-gon
3 sides – Triangle
4 sides – Quadrilateral
5 sides – Pentagon
6 sides – Hexagon
7 sides – Heptagon

3. Vocabulary
Every segment in the polygon that joins two non-consecutive vertices is a diagonal.
Interior angles are formed by two consecutive sides of a polygon. Exterior angles are adjacent and supplementary to an interior angle of the polygon.

4. Finding Sum of Angles
To find the number of degrees in a polygon, draw all the diagonals possible from one vertex. Then count the number of triangles formed and multiply that by 180 (the number of degrees in one triangle).
Example: When 2 diagonals are drawn in the figure below, 3 triangles are formed. In conclusion, the sum of the measures of the angles in a pentagon is 3(180) or 540°.
But, by using Theorem 55, the sum of the measures of the angles could be found in an easier way.

5. Theorem 55
The sum Si of the measures of the angles of a polygon with n sides is given by the formula Si = (n-2)180.
Example:
What is the sum of the measures of the angles in a heptagon?
Solution: Use the formula above and substitute for n
Si = (7-2)180
= (5)180
= 900°

6. Theorem 56
If one exterior angel is taken at each vertex, the sum Se of the measures of the exterior angles of a polygon is given by the formula Se = 360.
Therefore, the sum of the measures of the exterior angles in any polygon is 360°.

7. Theorem 57
The number of diagonals that can be drawn in a polygon of n sides is given by the formula
d = n(n-3) 2
Example:
How many diagonals can be drawn in an 18-gon?
Solution: Use the formula above and substitute 18 for n
d = 18(18-3)
= 270
= 135 diagonals

8. Regular Polygon Formulas
To find the measure of one angle of a regular polygon with n sides, use the following formula:
I = (n-2)180 n
Example:
What is the measure of one angle in a regular nonagon?
Solution: Use the formula above and substitute 9 for n
I = (9-2)180 9
= 1260
= 140°

9. Regular Polygon Formulas (Cont.)
To find the measure of one exterior angle of a regular polygon with n sides, use the following formula:
E = 360 n
Example:
What is the measure of one exterior angle of a regular octagon?
Solution: Use the formula above and substitute 8 for n 8
= 45°

10. Practice Problems
How many sides does a polygon have if the sum of the measures of its angles is 3240°?
What is the sum of the measures of the angles of a 31-gon?
Given: m A = 85°, m B = 115°, m C = 95°,
m D = 100°
Find : m E

11. Practice Problems (Cont.)
What is the sum of the measures of the exterior angles, one per vertex, of a decagon?
What is the name of a polygon with 65 diagonals?
How many diagonals does a 22-gon have?
What is the measure of one angle of a regular decagon?
What regular polygon has an angle measuring 150°?

12. Practice Problems (Cont.)
9. What regular polygon has an exterior angle measuring 6°?
What is the measure of one exterior angle of a regular octagon?
Answers on next slide

13. Answers to Practice Problems
20 sides
5220°
m E = 145°
360°
13-gon
209
144
Dodecagon
60-gon
45°

14. Works Cited
Rhoad, Richard, George Milauskas, and Robert Whipple. Geometry for Enjoyment and Challenge. Evanston, Illinois: McDougal, Littell & Company, 1991.
Habeeb, Danielle. “Diagonals in a Polygon.”
Geometry for Middle School Teachers Institute.
CPTM. 24 May 2008 <
uga.edu/tweb/cptm1/dhabeeb/diagonals/
diagonalsinapolygon.htm>.