Sec. 3-5Sec. 3-5The Polygon Angle-Sum The Polygon Angle-Sum

TheoremsTheorems

Objectives: Objectives: a)a) To classify PolygonsTo classify Polygonsb)b) To find the sums of the To find the sums of the

measures of the interior & measures of the interior & exterior exterior s of Polygons.s of Polygons.

Polygon:Polygon:

A closed plane figure.A closed plane figure. w/ at least 3 sides (segments)w/ at least 3 sides (segments) The sides only intersect at their The sides only intersect at their

endpointsendpoints Name it by starting at a vertex & go Name it by starting at a vertex & go

around the figure clockwise or around the figure clockwise or counterclockwise listing each vertex counterclockwise listing each vertex you come across.you come across.

Which of the following figures are Which of the following figures are polygons?polygons?

yes No No

Example 1: Name the 3 polygons

S T

U

VW

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

NameName

Triangle

Quadrilateral

Pentagon

Hexagon

Heptagon

Octagon

Nonagon

Decagon

Dodecagon

N-gon

Interior Interior SumSum

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) 180S = (n -2) 180

Examples 2 & 3:Examples 2 & 3:

Find the interior Find the interior sum of a 15 – gon.sum of a 15 – gon.

S = (n – 2)180S = (n – 2)180

S = (15 – 2)180S = (15 – 2)180

S = (13)180S = (13)180

S = 2340S = 2340

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

S = (n – 2)180S = (n – 2)180

900 = (n – 2)180900 = (n – 2)180

5 = n – 2 5 = n – 2

n = 7 sides n = 7 sides

Special Polygons:Special Polygons:

Equilateral PolygonEquilateral Polygon – All sides are – All sides are ..

Equiangular PolygonEquiangular Polygon – All – All s are s are ..

Regular PolygonRegular Polygon – Both Equilateral & – Both Equilateral & Equiangular.Equiangular.

IV. Exterior s of a polygon.

1

23 1

2

3

4 5

Th(3-10) Polygon Exterior Th(3-10) Polygon Exterior -Sum -Sum ThmThm The sum of the The sum of the

measures of the measures of the exterior exterior s of a s of a polygon is 360°.polygon is 360°.

Only one exterior Only one exterior per vertex. per vertex.

1

2

3

4 5mm1 + m1 + m2 + m2 + m3 + m3 + m4 + 4 + mm5 = 3605 = 360

For Regular Polygons

360

n= measure of one exterior

The interior & the exterior are Supplementary. IntInt + Ext + Ext = = 180180

Example 4: Example 4:

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

= 36

360 = 36n

10 = n

Example 5:Example 5: Find the sum of the interior Find the sum of the interior s of a s of a

polygon if it has one exterior polygon if it has one exterior measure measure of 24.of 24.

360

n= 24

n = 15

S = (n - 2)180

= (15 – 2)180

= (13)180

= 2340

Example 6:Example 6:

Solve for x in the following example.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: Example 7:

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

S = (n – 2)180S = (n – 2)180

= (6 – 2)180= (6 – 2)180

= (4)180= (4)180

= 720= 720

720

6

= 120