Geometry

11.4 Circumference and Arc Length

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Goals

Find the circumference of a circle. Find the length of an arc. Solve problems using circumference

and arc length. Conversions involving radians

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Circumference

The circumference is the distance around a circle.

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Circumference

You can find the circumference if you know the diameter (d) or the radius (r) of the circle.

2

C d

C r

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Example

Find the circumference of a circle with a radius of 8.

2 (8)

16

50.24

C

This is the circumference “in terms of pi”.

When computing C in decimal, use 3.14 or the pi button.

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Semicircle

Half of a circle. Its measure is 180. Its length is one-half the

circumference.

Converting Degrees to Radians

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Practice

A circle is 360 degrees. How many radians does that equal?

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Practice

A circle is 360 degrees. How many radians does that equal?

Think about when we inscribed a hexagon in a circle.

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How many radians are in a semicircle?

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Radians to Degrees

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Practice

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Practice

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Arc Length

The ratio of the length of an arc to the circumference of a circle is equal to the ratio of the measure of the arc to 360.

A

B

Arc Length of AB2 360

mABr

Recall: the measure of the arc is equal to the central angle that intercepts it.

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Example Find the length of AB.

A

B5

80L

802 (5) 360L

Write the proportion:

Length of Arc

Circumference of circle

Central Angle

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Solve the proportion

802 (5) 360

210 9

9 20

206.98

9

L

L

L

L

A

B5

80L6.98

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Your Turn Find the length of AB

A

B8

25

252 (8) 360

2516 360

360 400

109

3.43

9

40060

x

x

x

x

x

x

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Example

The length of AB is 25 and the radius is 10. Find the mAB.

A

B

10

x25

252 (10) 360

20 9000

9000 45020

143.2

x

x

x

x

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Problem 1

An indoor track at a gym has the design below. What is the length of the entire track?

100 ft

40 ft

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Problem 1 Solution

The ends make one circle.

100 ft

40 ft

The diameter is 40 ft.

C = (40)

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Problem 1 Solution

100 ft

40 ft

Add 200 for the straight segments.

Perimeter = 40 + 200 ft

Or, 325.66 ft.

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Summary

The length of an arc on a circle is proportional to the measure of the arc and to the central angle.

A semicircle has 180 and it’s length is half the circumference.