Area of a Circle Examples in Math

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

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 amount of space enclosed inside a circle, calculated as $\pi$ times the square of the radius.

Imagine cutting a pizza into many thin slices and rearranging them into a shape that looks like a rectangle. The 'height' of that rectangle is the radius $r$ , and the 'width' is half the circumference ( $\pi r$ ). So the area is $r \times \pi r = \pi r^2$ .

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: Area of a circle is the flat space inside it, $\pi r^2$ .

Common stuck point: The procedure for area of a circle is the easy part; the trap is plugging in the diameter for $r$ . Asking "Am I measuring the flat space inside a circle, not the length around it?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I measuring the flat space inside a circle, not the length around it?

Worked Examples

Example 1

easy

Find the area of a circle with radius

6

cm. Leave your answer in terms of

\pi

Find the area of the shaded circle.

Answer

A = 36\pi \text{ cm}^2

First step

The area enclosed by a circle of radius

r

A = \pi r^2

. This can be understood by imagining the circle divided into many thin triangles from the centre; their combined area gives

\frac{1}{2} \times (2\pi r) \times r = \pi r^2

Full solution

2
Substitute $r = 6$ cm: $A = \pi(6)^2 = \pi \times 36$ .
3
Result: $A = 36\pi$ cm² $\approx 113.1$ cm². Note that doubling the radius quadruples the area (since $r$ is squared), a key scaling insight.

The area of a circle depends on the square of the radius. Doubling the radius quadruples the area, which illustrates the quadratic relationship between radius and area.

Example 2

medium

Find the area of a semicircle with diameter

20

cm. Give your answer to one decimal place.

Find the area of the semicircle (diameter = 20 cm).

Example 3

easy

Show step by step how to find the area of a circle with radius

10

m, leaving the answer in terms of

\pi

Find the area in terms of π (show steps).

Example 4

medium

60°

sector of a circle with radius

6

— find its area in terms of

\pi

60° sector with r = 6. Find its area in terms of π.

Example 5

medium

A circular running track has radius

50

m. Find its area in square metres, using

\pi \approx 3.14

Example 6

hard

Two circles have radii

5

and

12

. A single circle has area equal to the sum of theirs. Find its radius.

Practice Problems

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

Example 1

hard

A circular garden has an area of

154

m². Find the radius of the garden. Use

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

A = 154 m². Find the radius r.

Example 2

medium

A circular pond has radius

6

m. A path around it extends the outer radius to

9

m. Find the area of the path.

Example 3

easy

Find the area of a circle with radius

3

. (Leave answer in terms of

\pi

Find the area (leave in terms of π).

Example 4

easy

Find the area of a circle with radius

5

. (In terms of

\pi

Find the area (leave in terms of π).

Example 5

easy

A circle has diameter

10

. Find its area in terms of

\pi

Diameter = 10. Find the area.

Example 6

easy

Find the area of a circle with radius

2

, using

\pi \approx 3.14

Example 7

easy

A circle has area

16\pi

. Find its radius.

Area = 16π. Find r.

Example 8

easy

Find the circumference of a circle with radius

7

. (In terms of

\pi

Example 9

easy

A circle has radius

1

. What is its area?

Example 10

easy

If a circle's radius doubles, what happens to its area?

Example 11

medium

Find the area of a semicircle with radius

6

. (In terms of

\pi

Semicircle with r = 6. Find area in terms of π.

Example 12

medium

A pizza has radius

8

inches. What is its area? (Use

\pi \approx 3.14

Pizza with r = 8 in. Find the area using π ≈ 3.14.

Example 13

medium

Find the area of a quarter-circle with radius

4

. (In terms of

\pi

Quarter-circle with r = 4. Find the area in terms of π.

Example 14

medium

A circular garden has area

49\pi

m². What is its circumference? (In terms of

\pi

Example 15

medium

A square has a circle inscribed in it. The square has side

10

. Find the circle's area. (In terms of

\pi

Example 16

medium

Find the area between two concentric circles (an annulus) with radii

3

and

5

. (In terms of

\pi

Example 17

medium

A circle is inscribed in a square of area

36

. What is the circle's area? (In terms of

\pi

Example 18

medium

Find the area of a circle with circumference

12\pi

. (In terms of

\pi

Example 19

medium

90°

sector of a circle with radius

4

— find its area. (In terms of

\pi

90° sector with r = 4. Find the sector area in terms of π.

Example 20

challenge

A circle is inscribed in an equilateral triangle of side

6

. (Skip exact — set up.) What's the radius of the inscribed circle?

Example 21

challenge

Two circles have radii

3

and

4

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

Example 22

challenge

A circle of radius

r

has the same area as a square of side

s

. Express

s

in terms of

r

Example 23

easy

Find the area of a circle with radius

4

. Leave your answer in terms of

\pi

Find the area in terms of π.

Example 24

easy

A circle has radius

9

. What is its area in terms of

\pi

Find the area in terms of π.

Example 25

easy

A circle has diameter

14

cm. Find its area in terms of

\pi

Diameter = 14 cm. Find the area in terms of π.

Example 26

easy

Estimate the area of a circle with radius

5

, using

\pi \approx 3.14

Example 27

medium

A circular tabletop has radius

0.6

m. Find its area to two decimal places, using

\pi \approx 3.14

Example 28

medium

A semicircle has diameter

12

. Find its exact area in terms of

\pi

Semicircle with diameter = 12. Find the exact area in terms of π.

Example 29

medium

Find the area of an annulus with outer radius

10

and inner radius

6

, in terms of

\pi

Example 30

medium

A circle's circumference is

20\pi

. Find its area in terms of

\pi

Example 31

medium

A pizza of diameter

16

inches is cut into

8

equal slices. What is the area of one slice in terms of

\pi

Pizza diameter = 16 in, cut into 8 slices. Find the area of one slice.

Example 32

medium

A circle is inscribed in a square whose perimeter is

40

. Find the area of the circle in terms of

\pi

Example 33

medium

A quarter circle has radius

8

. Find its area in terms of

\pi

Quarter-circle with r = 8. Find the area in terms of π.

Example 34

hard

A circle is inscribed in a square of side

12

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

\pi

Example 35

hard

A square is inscribed in a circle of radius

5

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

\pi

Example 36

hard

A circular pizza of radius

12

in is cut by two perpendicular diameters into four equal pieces. Find the area of one piece in terms of

\pi

Pizza r = 12 in cut into 4 equal pieces. Find one piece's area.

Example 37

hard

A circular lawn sprinkler covers a quarter-circle with radius

20

ft. What area does it water, in terms of

\pi

Sprinkler covers a 90° sector with r = 20 ft. Find the area.

Example 38

challenge

A circle has the same area as an equilateral triangle of side

6

. Find the circle's radius exactly.

Example 39

challenge

A circle of radius

r

is inscribed in a square. What fraction of the square's area is occupied by the circle? Give the exact value.

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Related Concepts

Circles Pi (π)Area Sector Area Volume of a Cylinder Surface Area of a Cylinder

Background Knowledge

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

circlespiarea