Geometry

10.5 Areas of Circles and Polygons

ObjectivesFind the area of a circle and polygonsTo be able to solve problems with circles

and polygons

Area of a CircleThe area of a circle is times the square of

the radius or A = r2.

rr

Using the Area of a CircleFind the area of

P.Solution:a. Use r = 8 in the

area formula.A = r2 = • 82

= 64 201.06

8 in.

P

So, the area if 64, or about 201.06 square inches.

Using the Area of a CircleFind the diameter

of Z.

Solution:Area of circle Z is 96

cm2. A = r2 96= r2 96/ = r2

30.56 r2

5.53 r

The diameter of the circle is about 11.06 cm.

Z

The Apothem

The apothem (a) is the segment drawn from the center of the polygon to the midpoint of the side (and perpendicular to the side)

aaa

More . . .

a

G

F

E

D C

B

A

H

Hexagon ABCDEF with

•center G

•radius GA

• apothem GH

Area of a regular polygonArea of a regular polygon

The area of a regular polygon is: The area of a regular polygon is:

A = ½ PaA = ½ PaArea Area

Perimeter Perimeter

apothemapothem

45-45-90

1. Perimeter = 14+14+14 = 42 m

2. Apothem: “30-60-90” triangle = (7√3)/3 = 4.0

3. Area = ½ Pa = ½ * 14 * 3 * 4 = 84 m²

1. Perimeter = 15*4= 60 in

2. Apothem: “45-45-90” triangle = 7.5 in

3. Area = ½ Pa = ½ * 4*15*7.5 = 225 in²

• Perimeter = 10*5= 50 m

• Apothem: “36-54-90” triangle = 5/tan 36 = 6.9

• Area = ½ Pa = ½ * 50* 6.9 = 172.5 m²

1. Find central angle measure : 360/n = 360/ 5 = 72°

2. Apothem splits angle into halves 72/2 = 36 °

A = ½ PaA = ½ Pa

If a = 5√3 ,

than we have

30-60-90 triangle

so the side = 10

A = ½*6*10*5 √3 = 259.8cm²

A = ½ PaA = ½ Pa

If a = 7.5 ,

and P = 54.49

A = ½*7.5 * 54.49 = 204.3 m²

A region = A square – A circleA region = A square – A circle

A square = 40*40 = 1600A square = 40*40 = 1600

A circle = 3.14*20² = 1256A circle = 3.14*20² = 1256

A region = 1600-1256 = 344 m² A region = 1600-1256 = 344 m²

A region = A circle – A (2 smaller circles)A region = A circle – A (2 smaller circles)

A circle= 64*3.14 = 200.96 A circle= 64*3.14 = 200.96

A one smaller circle = 16*3.14 = 50.24A one smaller circle = 16*3.14 = 50.24

A region = 200.96-2*50.24=100.5m² A region = 200.96-2*50.24=100.5m²