The Polygon Angle-Sum

Theorems

Objectives:

a) To classify Polygons

b) To find the sums of the measures of the interior & exterior s of Polygons.

Polygon:

• A closed plane figure.

• w/ at least 3 sides (segments)

• The sides only intersect at their endpoints

• Name it by starting at a vertex & go around the figure clockwise or counterclockwise listing each vertex you come across.

Which of the following figures are polygons?

yes No No

Example 1: Name the 3 polygons

S T

U

V W

X

Top

XSTU

Bottom

WVUX

Big

STUVWX

I. Classify Polygons by the number of sides it has.

Sides

3

4

5

6

7

8

9

10

12

n

Name

Triangle

Quadrilateral

Pentagon

Hexagon

Heptagon

Octagon

Nonagon

Decagon

Dodecagon

N-gon

Interior Sum

II. Also classify polygons by their Shape

a) Convex Polygon – Has no diagonal w/ points

outside the polygon.

E A

B

C

D

b) Concave Polygon – Has at least one diagonal w/

points outside the polygon.

* All polygons are convex

unless stated otherwise.

III. Polygon Interior sum

4 sides

2 Δs

2 • 180 = 360

5 sides

3 Δs

3 • 180 = 540

6 sides

4 Δs

4 • 180 = 720

• All interior sums are

multiple of 180°

Th(3-9) Polygon Angle – Sum Thm

Sum of Interior

# of sides

S = (n -2) 180

Examples 2 & 3:

• Find the interior sum of a 15 – gon.

S = (n – 2)180

S = (15 – 2)180

S = (13)180

S = 2340

• Find the number of sides of a polygon if it has an sum of 900°.

S = (n – 2)180

900 = (n – 2)180

5 = n – 2

n = 7 sides

Special Polygons:

• Equilateral Polygon – All sides are .

• Equiangular Polygon – All s are .

• Regular Polygon – Both Equilateral & Equiangular.

IV. Exterior s of a polygon.

1

2 3 1

2

3

4 5

Polygon Exterior -Sum Thm • The sum of the measures of

the exterior s of a polygon is 360°.

» Only one exterior per vertex.

360

n

1

2

3

4 5 m1 + m2 + m3 + m4 + m5 = 360

For Regular Polygons

= measure of one

exterior

The interior & the exterior

are Supplementary.

Int + Ext = 180

Example 4:

• How many sides does a polygon have if it has an exterior measure of 36°.

= 36

360 = 36n

10 = n

Example 5:

• Find the sum of the interior s of a polygon if it has one exterior measure of 24.

360

n

= 24

n = 15

S = (n - 2)180

= (15 – 2)180

= (13)180

= 2340

Example 6:

• Solve for x in the following example.

x

100

4 sides

Total sum of interior s = 360

90 + 90 + 100 + x = 360

280 + x = 360

x = 80

Example 7:

• Find the measure of one interior of a regular hexagon.

S = (n – 2)180

= (6 – 2)180

= (4)180

= 720

720

6

= 120