Formulas Involving Polygons
Chapter 7 Section 3
By: Alex Pipcho
Polygon Names
•
•
•
•
•
•
3 sides – Triangle
•
4 sides – Quadrilateral
•
5 sides – Pentagon
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6 sides – Hexagon
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7 sides – Heptagon
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8 sides – Octagon
9 sides- Nonagon
10 sides – Decagon
12 sides – Dodecagon
15 sides –Pentadecagon
n sides – n-gon
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.
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.
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
7 for n
Si = (7-2)180
= (5)180
= 900°
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°.
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)
2
= 270
2
= 135 diagonals
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
9
= 140°
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
E = 360
8
= 45°
Practice Problems
1. How many sides does a polygon have if the
sum of the measures of its angles is 3240°?
2. What is the sum of the measures of the angles
of a 31-gon?
3. Given: m A = 85°, m B = 115°, m C = 95°,
m D = 100°
Find : m E
Practice Problems (Cont.)
4. What is the sum of the measures of the
exterior angles, one per vertex, of a decagon?
5. What is the name of a polygon with 65
diagonals?
6. How many diagonals does a 22-gon have?
7. What is the measure of one angle of a regular
decagon?
8. What regular polygon has an angle measuring
150°?
Practice Problems (Cont.)
9. What regular polygon has an exterior
angle measuring 6°?
10. What is the measure of one exterior
angle of a regular octagon?
Answers on next slide
Answers to Practice Problems
1. 20 sides
2. 5220°
3. m E = 145°
4. 360°
5. 13-gon
6. 209
7. 144
8. Dodecagon
9. 60-gon
10. 45°
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 <http://intermath.coe.
uga.edu/tweb/cptm1/dhabeeb/diagonals/
diagonalsinapolygon.htm>.