# of sides

# of triangles

Sum of measures of

interior angles

3 1 1(180)=180

4 2 2(180)=360

5 3 3(180)=540

6 4 4(180)=720

n n-2 (n-2) • 180

If a convex polygon has n sides, then the sum of the measure of

the interior angles is

(n – 2)(180°)

If a regular convex polygon has n sides, then the measure of one of the interior angles is

n

n 180)2(

Ex. 1 Use a regular 15-gon to answer the questions.

A)Find the sum of the measures of the interior angles.

B) Find the measure of ONE interior angle

2340°

156°

Ex: 2 Find the value of x in the polygon

130

126

143

100

117

x

126 + 130 + 117 + 143 + 100 + x = 720

616 + x = 720

x = 104

Ex: 3 The measure of each interior angle is 150°, how many sides does the regular polygon have?

n

n 180)2(One interior angle

150180)2(

n

n

nn 150180)2(

nn 150360180 36030 n

12nA regular dodecagon

Two more important terms

Exterior Angles

Interior Angles

The sum of the measures of the

exterior angles of a convex polygon, one

at each vertex, is 360°.

1

2

3

4

5

m m m m m 1 2 3 4 5 360

1

3

2

m m m 1 2 3 360

The sum of the measures of the

exterior angles of a convex polygon, one

at each vertex, is 360°.

1

3

2

4

m m m m 1 2 3 4 360

The sum of the measures of the

exterior angles of a convex polygon, one

at each vertex, is 360°.

The measure of each exterior angle of a regular polygon is

n

360

Ex. 4 Find the measure of ONE exterior angle of a regular 20-gon.

18°

sum of the exterior anglesnumber of sides

360

20

Ex. 5 Find the measure of ONE exterior angle of a regular heptagon.

51.4°

sum of the exterior anglesnumber of sides

3607

Ex. 6 The sum of the measures of five interior angles of a hexagon is 625. What is the measure of the sixth angle?

95°

Let’s practice!

11.1 Worksheet

s s

Area of an Equilateral Triangle

213

4A s

s

6060

30 30

Ex: 1 Find the area of an equilateral triangle with 4 ft sides.

213

4A s 24325.A

293.6 ftA

A Circle can be circumscribed around any regular polygon

VERTICES

A RADIUS joins the center of the regular polygon with any of the

vertices

A Central Angle is an angle whose vertex is the center and whose sides are two consecutive radii

n

360

A Regular HexagonEqual Angles

Equal Sides

How many equilateral triangles make up a regular Hexagon?

What is the area of each triangle?

What is the area of the hexagon?

s

213

4A s

6 • (the area of the triangle)

4

213

4A s The area of an equilateral triangle

The area of our equilateral triangle in this exampleA = 6.9282

The area of our hexagon in this exampleA = 6 * (6.9282)

How many identical equilateral triangles do we have? 6

41.569 units2 What is the area of this regular hexagon?

An APOTHEM is the distance between the center and a side. (It MUST be perpendicular to the side.)

How to find the

You need to know the apothem and perimeter

Area = (1/2)•a•P

or A = .5•a•P

Area of a Regular Polygon:

A = ½ aP

a

A = .5 (apothem) (# of sides)(length of each side)

A Regular Octagon

7 ft

45

7

3.5 ft

22.5°

x

360/8=45

7 ft

Perimeter is 56 feetApothem is 8.45 feet

What is the area?

Area = .5 • 8.45 • 56

Area = 236.6 ft2

11.2 WorksheetPractice B

ODDS

Worksheets’ EVENS