11.1 NOTES ­ Circumference and Arc Length

1

BELLWORK: What is π?

3.14159....

The ratio of a circle's circumference to its diameter.

LESSON 11.1 - Circumference and Arc Length

• The CIRCUMFERENCE of a circle is the distance around the circle.

• For all circles, the ratio of the circumference C to the diameter d is the same.  This ratio is        = π .

• Solving the ratio for C yields the formula for the circumference of a circle.

CIRCUMFERENCE OF A CIRCLE

Cd

C = πdOR

C = 2πr

rr

d

Find the circumference of a circle with a diameter of 12 inches. Round to the nearest tenth.

12 in

C = 12π

C ≈ 37.7 in

Find the circumference of a circle with a radius of 93 centimeters. Round to the nearest tenth.

93 cm

C = 2 π  93

C ≈ 584.3 cm

Find the diameter of a circle with a circumference of 62 feet. Round to the nearest tenth.

d

62 = dπ

d ≈ 19.7 ft

11.1 NOTES ­ Circumference and Arc Length

2

Find the radius of a circle with a circumference of 7 millimeters. Round to the nearest tenth.

r7 = 2 π  r

r ≈ 1.1 mm

The measurements of a tire are given below. How many FEET does the tire travel in one revolution? If the tire has traveled 500 feet, how many revolutions did the tire complete?

17 inches

5 inches

5 inches

d = 5 + 17 + 5

d = 27 inches

C = 27πC ≈ 84.8 in

C ≈ 7.1 ft

The tire covers about 7.1 feet in

one revolution

500 = x(27π)x ≈ 5.9 rev.

In order to travel 500 feet, the tire must make almost 6 revolutions (5.9)

• An ARC LENGTH is a portion of the circumference of a circle.  

• We can use the measure of an arc (in degrees) to find its length (in linear units, such as centimeters).

ARC LENGTHIn a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360°.

C

A

B

Arc length of AB2πr

mAB360°

=

Find the length of arc AB given that the radius is 11 cm and mAB = 64°.

11 cm

64° C

A

B

AB 64°2 π 11 360°

=

AB22π = 0.18

AB = 12.3 cm

Find the length of arc AB given that the radius is 53 ft and mAB = 165°.

53 ft

165°

CA

B

AB 165°2 π 53 360°

=

AB106π = 0.4583

AB = 152.6 ft

Find the length of a radius of C given that mAB = 182° and the length of AB is 63 yards.

63 yd

182° C

A

B63 182°2 π r 360°=

632π  r = 0.5056

r = 19.8 yd

10.02r = 0.5056

10.02 = 0.5056r

11.1 NOTES ­ Circumference and Arc Length

3

Find the circumference of C given that mAB = 76° and the length of AB is 14 meters.

14 m

76°C

AB14 76°

2 π r 360°=

142π  r = 0.2111

r = 10.6 m

2.23r = 0.2111

2.23 = 0.2111r

Find mAB given that the radius of C is 29 miles and the length of AB is 56 miles.

56 mi

29 miC

A

B

56 m AB2 π 29 360°

=

m AB360°

0.307 =

m AB = 110.6°

Find mAB given that the radius of C is 44 feet and the length of AB is 32 feet.

32 ft

44 ft

C

A

B

32 m AB2 π 44 360°

=

0.116 =

m AB = 41.7°

m AB360°

HOMEWORK:11.1 Worksheet ‑ Circumference and Arc Length