Circumference Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Circumference.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The total distance around the outside of a circle; equal to $\pi$ times the diameter or $2\pi r$ .

Imagine wrapping a string tightly around a circular jar lid, then straightening the string out. That length is the circumference. No matter the size of the circle, the circumference is always $\pi$ times the diameter—roughly $3.14$ laps of the diameter around the edge.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Circumference is the perimeter of a circle, always $\pi$ times the diameter.

Common stuck point: The procedure for circumference is the easy part; the trap is squaring the radius. Asking "Am I measuring the length around a circle's edge, not the space inside?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I measuring the length around a circle's edge, not the space inside?

Worked Examples

Example 1

easy

Find the circumference of a circle with radius

5

cm. Leave your answer in terms of

\pi

Circle with radius 5 cm; find the circumference.

Answer

C = 10\pi \text{ cm}

First step

The circumference is the perimeter of a circle — the distance around it. Two equivalent formulas:

C = 2\pi r

(using radius) or

C = \pi d

(using diameter). They are equivalent since

d = 2r

Full solution

2
Substitute $r = 5$ cm into $C = 2\pi r$ : $C = 2\pi(5) = 10\pi$ .
3
Result: $C = 10\pi$ cm $\approx 31.4$ cm. The formula $C = 2\pi r$ encodes the definition of $\pi$ itself: $\pi = C/d$ , the ratio of circumference to diameter, which is the same for every circle.

The circumference is the distance around a circle. The constant

\pi \approx 3.14159

is the ratio of any circle's circumference to its diameter, making

C = \pi d = 2\pi r

Example 2

medium

A circular track has a circumference of

400

m. Find the radius of the track to the nearest metre.

Example 3

medium

A bicycle wheel of radius

35

cm makes

50

revolutions. Find the total distance traveled. Use

\pi\approx 22/7

Example 4

hard

A car tire has diameter

0.7

m. Use

\pi\approx 22/7

to find how many rotations are needed to travel

1100

Example 5

hard

Two pulleys of radius

5

are

20

apart. Find the belt length using

\pi\approx 3.14

Example 6

challenge

Adding

2\pi

meters to the equator's wrapped string and lifting uniformly off Earth raises it by how much?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find the circumference of a circle with diameter

14

cm. Leave your answer in terms of

\pi

Circle with diameter 14 cm; use C = πd.

Example 2

medium

A bicycle wheel has diameter

70

cm. How far does it travel in

20

complete rotations? Use

\pi = \frac{22}{7}

Example 3

easy

A circle has radius

7

. Find its circumference in terms of

\pi

Circle with radius 7; find circumference.

Example 4

easy

A circle has diameter

22

. Find its circumference in terms of

\pi

Circle with diameter 22; use C = πd.

Example 5

easy

A circle has circumference

30\pi

. Find its radius.

Circumference = 30π; find radius r.

Example 6

easy

A circle has circumference

44

. Use

\pi\approx 22/7

to find its diameter.

Example 7

easy

A circular pond has diameter

8

m. Use

\pi\approx 3.14

to estimate the distance walked around it once.

Example 8

medium

Find the arc length of a

120^\circ

sector in a circle of radius

9

in terms of

\pi

120° sector of circle with radius 9; find the arc length.

Example 9

medium

Find the perimeter of a semicircle with radius

4

(including the diameter).

Semicircle with radius 4; total perimeter = arc + diameter.

Example 10

medium

Two circles have circumferences

20\pi

and

50\pi

. Find the ratio of their radii.

Example 11

medium

A clock's minute hand is

7

cm long. How far does its tip travel in one hour? Use

\pi\approx 22/7

Example 12

medium

A wire of length

66

cm is bent into a circle. Find its radius using

\pi\approx 22/7

Example 13

medium

Find the arc length of a

45^\circ

sector in a circle of radius

16

in terms of

\pi

45° sector of circle with radius 16; find the arc length.

Example 14

hard

A circle is inscribed in a square of side

14

. Find the difference between the square's perimeter and the circle's circumference. Use

\pi\approx 22/7

Example 15

hard

A regular hexagon has perimeter

48

. Find the circumference of its circumscribed circle in terms of

\pi

Example 16

hard

A semicircular window has straight edge

10

ft. Find its total perimeter using

\pi\approx 3.14

Example 17

hard

Find the arc length corresponding to a central angle of

1

radian in a circle of radius

5

Example 18

hard

A circular running track has circumference

400

m. A runner completes

5

laps. Find the distance.

Example 19

challenge

A circular sector has arc length

8\pi

and central angle

240^\circ

. Find the radius.

240° sector with arc length 8π; find radius r.

Example 20

challenge

A wheel rolls one full revolution without slipping. The center moves how far in terms of the radius

r

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Related Concepts

Circles Pi (π)Perimeter Arc Length Surface Area of a Cylinder Area of a Circle

Background Knowledge

These ideas may be useful before you work through the harder examples.

circlespiperimeter