Area of Regular Polygons11.2

Chapter 11Areas of Polygons and Circles

Section 11.2

Area of Regular Polygons

Find the area of an equilateral triangle

Find the area of a regular polygon

Area of Regular Polygons11.2

Example 1 Area Equilateral Triangle

Example 2 Area Regular Pentagon

Example 3 Area Regular Octagon

Example 4 Always, Sometimes, Or Never

CA Standard 12.0

Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems. (Key)

Area of Regular Polygons11.2

Find the area of an equilateral triangle

12

A bh

60° 60°

30°

2s

2s

32s

1 32 2

sA s

2 34

sA

Area of Regular Polygons11.2

Find the area of an equilateral triangle

2 34

sA

25 34

25 34

Area of Regular Polygons11.2

Find the area of an equilateral triangle

Find the area of the triangle

The Triangle is equilateral with s = 11

2 34

sA Area Equilateral

211 34

A s = 11

121 34

A Simplify

Answer: The triangle has area of square units121 3

52.44

Area of Regular Polygons11.2

Find the area of an equilateral triangle

Find the area of the triangle

Answer: The triangle has area of square units5 3

2.24

5 5

5

Area of Regular Polygons11.2

Center of circumscribed circle

Radius of circumscribed circle

Distance from center to any side of the polygons

Vertex is on the center, Sides connect consecutive vertices

Area of Regular Polygons11.2

To find the area of the hexagon, find the area of one triangle and multiply by 6

12

6 bhA

Height of the triangle is an apothem of the hexagon

Base of the triangle is a side of the hexagon

6 is the number of sides of the hexagon

12ans

b

h

12

n as

Area of Regular Polygons11.2

Find the area of a regular polygon

5

4

Perimeter = ns = 5(5) = 25 units

1 1(4)(5)(5) 50

2 2Area ans square units

Area of Regular Polygons11.2

Find the area of a regular polygon

Need to find the apothem and side length

Apothem = BD

Side Length = AC

Look at ABC

Area of Regular Polygons11.2

1(360) 72

5m ABC

Find the area of a regular polygon

1(72) 36

2m DBC

Find BD and AC

BD = cos 36 AC = 2 sin 36

Area of Regular Polygons11.2

Find the area of a regular polygon

Apothem = BD = cos 36

s = AC = 2(DC) = 2 sin 36

12

A ans

1(cos36)(5)(2sin36)

2A

5(cos36)(sin36)A

2.4A square units

Area of Regular Polygons11.2

Find the area of a regular polygon

Find the area of an octagon with radius 3 cm

Need to find the apothem and side length

a

s

Central Angle has measure 360 8 = 45

This angle is half the

Central Angle

45 2 = 22.5

This length is half of s

2s

Area of Regular Polygons11.2

Find the area of a regular polygon

Find the area of an octagon with radius 3 cm

Need to find a and s using the triangle

cos 22.53a 3cos 22.5a

2sin 22.53

s

3sin 22.52s

s = 6 sin 22.5

a

2s

22.5°

Area of Regular Polygons11.2

Find the area of a regular polygon

Find the area of an octagon with radius 3 cm

a

s

2s

12

A ans

1(3cos22.5)(8)(6sin 22.5)

2

225.5cm

3cos 22.5a s = 6 sin 22.5

Area of Regular Polygons11.2

Find the Area and Perimeter of the following pentagon

Find the measures of interior and exterior angles of polygons

Answer: Area = 1125 tan 36 817.4 Square UnitsPerimeter = 150 tan 36 109 Units

Area of Regular Polygons11.2

Always

Always

Area of Regular Polygons11.2

HW # 48Pg 672-675 9-29, 30-52