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Geometry

Circumference and Arc Length

Lesson

The distance around a circle is the circumference. An arc length is a piece of it — proportional to the central angle.

Formulas

C=2πr=πdC = 2\pi r = \pi d
arc length=θ360∘⋅2πr\text{arc length} = \frac{\theta}{360^\circ} \cdot 2\pi r

Where θ\theta is the central angle in degrees.

Worked example

Arc of 90° in a circle with r=4r = 4, using π≈3.14\pi \approx 3.14:

arc=14⋅2⋅3.14⋅4=6.28\text{arc} = \tfrac{1}{4} \cdot 2 \cdot 3.14 \cdot 4 = 6.28

Use π≈3.14\pi \approx 3.14 for all problems in this topic.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

C for r=5 (π=3.14)C \text{ for } r = 5 \ (\pi = 3.14)

Problem 2

C for r=10C \text{ for } r = 10

Problem 3

C for d=20C \text{ for } d = 20

Problem 4

Arc length: 90∘,r=4\text{Arc length: } 90^\circ, r = 4

Practice

Standard problems matching the lesson.

Problem 5

C for r=8C \text{ for } r = 8

Problem 6

C for r=1C \text{ for } r = 1

Problem 7

C for d=14C \text{ for } d = 14

Problem 8

C for r=15C \text{ for } r = 15

Problem 9

Arc: 180∘,r=10\text{Arc: } 180^\circ, r = 10

Problem 10

Arc: 45∘,r=8\text{Arc: } 45^\circ, r = 8

Problem 11

Arc: 60∘,r=12\text{Arc: } 60^\circ, r = 12

Problem 12

Arc: 270∘,r=4\text{Arc: } 270^\circ, r = 4

Problem 13

Arc: 120∘,r=6\text{Arc: } 120^\circ, r = 6

Problem 14

Arc: 30∘,r=12\text{Arc: } 30^\circ, r = 12

Problem 15

Distance per revolution, r=14 in\text{Distance per revolution, } r = 14 \text{ in}

Problem 16

Semicircle length, r=20\text{Semicircle length, } r = 20

Problem 17

Pizza crust, r=6 in\text{Pizza crust, } r = 6 \text{ in}

Problem 18

Quarter circle, r=10\text{Quarter circle, } r = 10

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

Arc: 150∘,r=18\text{Arc: } 150^\circ, r = 18

Problem 20

C for r=50C \text{ for } r = 50

Problem 21

C=18.84. Find r.C = 18.84. \text{ Find } r.

Problem 22

Wheel travels 188.4 ft, r=3 ft. Revolutions?\text{Wheel travels 188.4 ft, } r = 3 \text{ ft. Revolutions?}

Problem 23

Half-circle, r=20\text{Half-circle, } r = 20

Problem 24

Find r given diameter 22\text{Find } r \text{ given diameter 22}

Problem 25

Semicircle arc length 31.4. Radius?\text{Semicircle arc length 31.4. Radius?}

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