Areas of Regular Polygons 11.2

California State Standards 8: Solve problems involving perimeter and area.10: Find area of polygons16: Perform constructions19: Use trigonometric functions

theorem

Area of an Equilateral Triangle

ss

s

214 3A s

1 12 2 3A s s

12 2 3sA s

informal proof

s

s

60o

30o

2s

2 3s

214 3A s

example

4

214 3A s

214 3 (4)A

14 (16) 3A

4 3A

6.8 sq. unitsA

definition

Center of a PolygonThe center of its circumscribed circle.

Radius of a PolygonThe radius of its circumscribed circle.

definition

Apothem of a PolygonThe distance from the center to any side of the polygon.

radius

apothem

definition

Central Angle of a Regular PolygonAn angle whose vertex is the center andwhose sides contain two consecutive vertices.

definition

The Measure of a Central Angle of a Regular Polygon

72o360

n360

5

theorem

Area of a Regular Polygon P is the perimeter

a is the apothem

12A aP

s

a s

ss

s

1. 2pentA aP

125A aP#

125 5A a s#

125 5A as#

12A as#

informal proof

s

as

ss

s

sin 22.54

b

example

A regular octagon is inscribed in a circle with aradius 4 units. Find the area of the octagon.

4

22.5o

central angle = 45o

ba

s1.5b

3s

12A aP

3

12A aP

44.4 sq unitsA 1

2 3.7 24A

3.7a

cos 22.54

a

example

A regular octagon is inscribed in a circle with aradius 4 units. Find the area of the octagon.

4

22.5o

3

a3.7

example

The bottom of a glass is a regular dodecagon(12 sides) with a side length of 1.2 cm and aradius of 2.3 cm. What is the area of the bottom of the glass?

The bottom of a glass is a regular dodecagon(12 sides) with a side length of 1.2 cm and aradius of 2.3 cm. What is the area of the bottom of the glass?

example

1.20.6

2.3

215.8 cmA2 2 22.3 0.6a

14.4P 12(1.2)P

2.2 a

12 2.2 14.1A

12A aP

Today’s assignment p 672: 10 – 32 even

Find the area of the triangle.Find the measure of a central angle of a regular polygon with the given number of sides.

Find the area of the inscribed regular polygon shown.

Find the perimeter and area of the regular polygon.

Do the rest at home.