Geometry Honors Section 5.3

Circumference and Area of Circles

While the distance around the outside of a polygon is known as

the ________, the distance around the outside of a circle is called the

____________.

perimeter

circumference

For any circle, the ratio of the circumference to the diameter, , is

the same. This ratio is approximately equal to

___________. We use the Greek letter ____ to represent this

irrational number. A fractional approximation is _____.

Cd

3.141592656

227

Once again, = , so

C = ____

or in terms of the radius

C = ______

Cd

d

2 r

Activity 2 on page 316 explains how the formula for the area of a

circle is derived.

A = ______ 2r

Example: Find the circumference and area of a circle with a diameter of 12. Give an exact answer and an answer rounded to the nearest 1000th.

2 6 12 37.699C 2 26 36 113.097A r

6r

Example: Find the area of a circle with a circumference of .18

rrC

218

2

9r

8192

2

A

rA

Example: Find the area of the shaded region. Give an exact answer and an answer rounded to the nearest 1000th.

circlesquare AA

22 rs 22 510

25100

460.21

A sector of a circle is the region bounded by two radii and the arc

joining there outer endpoints.

Example: Find the area of sector AQB.

21236060

24

As you can see from the previous example, the

area of a sector =

OR

2

360M r

2 360Area of Sector M

r

Example: A circle has a diameter of 30 feet. If the area of a sector in this circle has a measure of ft2, find the measure of arc determining this sector.

2

15r

2 360Area of Sector M

r

360152

2

M

3602252 M

02.3

720225

M

M

A similar formula can be used to find the length of an arc. Length of an arc =

OR

2360M r

2 360length of arc M

r

Example: A circle has a radius of 6 cm. If an arc has a measure of 800, find the length of the arc.

2 360length of arc M

r

36080

12

L

6.2

960360

L

L

Example: An arc has a measure of 300 and a length of inches. What is the radius of the circle in which this arc is found?

18

2 360length of arc M

r

36030

218

r

108648060

rr

Note: The “measure of an arc” and the “length of an arc” are not the same thing. The measure of an arc is given in _______ and refers to ___________________ The length of an arc is given in __________ and refers to _______________________

degreesa fraction of the circle.

in / cm / ftthe distance along the arc.

While degree is certainly the unit of angle measure that you are

most familiar with, another commonly used unit for measuring

angles is the radian.

A radian is a unit of angle measure

radius. thelength toin equal is arc dintercepte

whosecircle a ofcenter at the anglean toequal

radian 1

Since the circumference of any circle is equal to ______, then there must be _____ arcs of length ___ on any circle.

Thus, the radian measure of a full circle is _____. We also know the degree measure

of a full circle is _______. Therefore, _______________________

or

2 r

2

r2

360360 radians 2

180 radians

Example: Convert 72o to radians

72180x

72180 x

18072

x

52

x

Example: Convert radians to degrees

x8

180

x

8

180

x 5.22 x5.22