Pi (π) Examples: 46 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Pi (π).

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 ratio of a circle's circumference to its diameter, approximately $3.14159\ldots$

No matter how big or small the circle, circumference $\div$ diameter always equals $\pi$ .

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: $\pi$ is the fixed ratio of any circle's distance-around to its distance-across, about $3.14159$ .

Common stuck point: The procedure for pi (π) is the easy part; the trap is using $C=\pi r$ instead of $C=\pi d$ . Asking "Am I converting between a circle's radius/diameter and its circumference or area?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I converting between a circle's radius/diameter and its circumference or area?

Worked Examples

Example 1

easy

A circle has a diameter of 10 cm. What is its circumference? Use

\pi \approx 3.14

Circle with diameter 10 cm — find the circumference.

Answer

C = 31.4

First step

Step 1: The formula for circumference is

C = \pi d

, where

d

is the diameter.

Full solution

2
Step 2: Substitute the values: $C = 3.14 \times 10$ .
3
Step 3: Calculate: $C = 31.4$ cm.

Pi (

\pi

) is the ratio of a circle's circumference to its diameter — it is always approximately 3.14159, no matter the size of the circle. Multiplying the diameter by

\pi

gives the circumference.

Example 2

medium

A circle has a radius of 7 m. Find its area. Use

\pi \approx 3.14

Circle with radius 7 m — find the area.

Example 3

medium

A circular pool has a circumference of 31.4 m. Find its radius and area. Use

\pi \approx 3.14

Example 4

medium

A pizza has diameter

12

inches. Find its area using

\pi \approx 3.14

Pizza with diameter 12 in (radius 6 in) — find the area.

Example 5

medium

A semicircle has radius

5

cm. Find its area and perimeter in terms of

\pi

Semicircle with radius 5 cm — find the area and perimeter.

Example 6

medium

A circular pond has circumference

44

m. Find its area in m

^2

using

\pi \approx \frac{22}{7}

Example 7

hard

A square has the same perimeter as a circle of radius

r

. Show which figure has the larger area, and by what factor.

Example 8

hard

A sector of a circle has central angle

60^\circ

and radius

9

cm. Find its area in terms of

\pi

Sector with central angle 60° and radius 9 cm — find the area.

Example 9

hard

Two circles have radii

3

and

4

. A third circle has area equal to the sum of their areas. Find its radius.

Example 10

challenge

A circle is inscribed in an equilateral triangle of side

6

cm. Find the area of the circle in terms of

\pi

Practice Problems

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

Example 1

easy

A circle's circumference is

C = 62.8

cm. What is its diameter? Use

\pi \approx 3.14

Example 2

hard

A wheel of radius 0.5 m rolls without slipping. How many full rotations does it make to travel 100 m? Use

\pi \approx 3.14159

Example 3

easy

\pi

is the ratio of a circle's circumference to which other measurement?

Example 4

easy

\pi

a rational or irrational number?

Example 5

easy

A circle has diameter

7

. Find its circumference in terms of

\pi

Example 6

easy

Which is the better approximation of

\pi

3.14

3

Example 7

easy

A circle has radius

5

. Find its area in terms of

\pi

Circle with radius 5 — find the area in terms of π.

Example 8

easy

If a circle's diameter doubles, by what factor does its circumference grow?

Example 9

easy

Estimate the area of a circle with radius

10

using

\pi \approx 3.14

Example 10

easy

True or false: the ratio

C/d

is larger for a bigger circle.

Example 11

medium

A circle has circumference

31.4

. Using

\pi \approx 3.14

, find its radius.

Circle with circumference 31.4 — find the radius.

Example 12

medium

Why is the fraction

\frac{22}{7}

only an approximation of

\pi

, not its exact value?

Example 13

medium

A circular track has radius

50

m. A runner completes 4 laps. About how far did they run? Use

\pi \approx 3.14

Example 14

medium

A circle's area is

36\pi

. Find its circumference in terms of

\pi

Example 15

medium

A pizza of diameter

16

inches is cut into 8 equal slices. Find the area of one slice in terms of

\pi

Pizza with radius 8 in cut into 8 equal slices (45° each) — find one slice's area.

Example 16

medium

A square and a circle have the same perimeter/circumference of

4\pi

. Which has the larger area?

Example 17

medium

How many times larger is the circumference of a circle than its diameter?

Example 18

medium

A circular garden of radius

r

is surrounded by a path. The outer edge of the path has radius

r + 1

. Express the path's area in terms of

\pi

and

r

Example 19

challenge

Ancient mathematicians estimated

\pi

by inscribing regular polygons in a circle. Explain why a regular polygon's perimeter, divided by the circle's diameter, approaches

\pi

as the number of sides increases.

Example 20

challenge

A rope is wrapped tightly around the Earth's equator (radius

R

). You then add just

2\pi

meters of extra rope and lift it to a uniform height above the surface. How high off the ground is the rope?

Example 21

challenge

A circle is inscribed in a square, and the square is inscribed in a larger circle. If the small circle has area

\pi

, find the area of the large circle in terms of

\pi

Example 22

challenge

Two pulleys of radius

3

are connected by a tight belt, their centers

10

apart. Find the total length of the belt in terms of

\pi

Example 23

easy

A circle has radius

4

cm. Find its circumference in terms of

\pi

Circle with radius 4 cm — find the circumference in terms of π.

Example 24

easy

A circle has diameter

10

cm. Find its area in terms of

\pi

Example 25

easy

A circle has circumference

20\pi

m. Find its radius.

Example 26

easy

A circle has radius

6

cm. Find its area in terms of

\pi

Example 27

easy

What is

\pi

in degrees of arc? (i.e.,

\pi

radians = ? degrees)

Example 28

medium

A bicycle wheel has diameter

0.7

m. How far does it travel in one full rotation? Use

\pi \approx 3.14

Example 29

medium

A circular garden has area

50.24

^2

. Find its radius using

\pi \approx 3.14

Example 30

medium

A circle has area

36\pi

^2

. Find its circumference in terms of

\pi

Example 31

medium

A clock's minute hand is

10

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

\pi \approx 3.14

Example 32

medium

A circle has circumference

C = 18\pi

cm. Find its area in terms of

\pi

Example 33

hard

A running track is a rectangle

80

m by

50

m with semicircles capping the short ends. Find the total distance around the track in terms of

\pi

Example 34

hard

A circle is inscribed in a square of side

10

cm. Find the area of the region inside the square but outside the circle, in terms of

\pi

Example 35

hard

A circle has area

A

and circumference

C

satisfying

A = C

. Find the radius.

Example 36

hard

A pendulum of length

2

m swings through an arc whose central angle is

30^\circ

. Find the arc length in terms of

\pi

Pendulum of length 2 m swinging through 30° — find the arc length.

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Related Concepts

Circles Division Circumference Circles Radians

Background Knowledge

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

circlesdivision