Circles Examples in Math

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

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 set of all points in a plane at a fixed distance (the radius) from a central point called the center.

Spin around with your arm fully outstretched—your fingertip traces a perfect circle.

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: A circle is every point that sits exactly the radius away from one central point.

Common stuck point: The procedure for circles is the easy part; the trap is mixing up radius and diameter. Asking "Is every point on the boundary the same distance from one center?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is every point on the boundary the same distance from one center?

Worked Examples

Example 1

easy

A circle has a radius of

7

cm. Find its diameter and state the relationship between the radius and the diameter.

Radius = 7 cm — find the diameter

Answer

d = 14 \text{ cm}

First step

Key circle relationships: the diameter

d

spans the full width through the centre, so

d = 2r

. The radius

r

is the distance from centre to any point on the circle, so

r = \frac{d}{2}

Full solution

2
Substitute $r = 7$ cm into $d = 2r$ : $d = 2(7) = 14$ cm.
3
Verify the relationship: $r = \frac{d}{2} = \frac{14}{2} = 7$ cm ✓. The diameter is always exactly twice the radius regardless of the circle's size.

The radius extends from the centre to any point on the circle, while the diameter passes through the centre connecting two opposite points. Understanding this relationship is fundamental to all circle calculations.

Example 2

medium

A chord of a circle is

24

cm long and is

5

cm from the centre. Find the radius of the circle.

Example 3

medium

A chord of length

16

6

from the center. Find the radius.

Chord = 16, distance from center = 6 — find the radius

Example 4

medium

A square has side

12

. Find the radius of the inscribed circle.

Example 5

hard

A chord of length

24

is in a circle of radius

13

. Find the chord's distance from the center.

Radius = 13, chord = 24 — find the distance from the center

Example 6

hard

Write the equation of a circle with center

(0,0)

and radius

7

Practice Problems

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

Example 1

easy

A circle has a diameter of

20

cm. What is its radius?

Diameter = 20 cm — find the radius

Example 2

easy

Two radii and one chord form a triangle inside a circle. If the radius is

10

cm and the triangle's perimeter is

34

cm, find the chord length.

Example 3

easy

A circle has radius

5

. What is its diameter?

Radius = 5 — find the diameter

Example 4

easy

A circle has diameter

14

. What is its radius?

Diameter = 14 — find the radius

Example 5

easy

What is the name for the distance from the center of a circle to any point on the circle?

Example 6

easy

Which is longer in a given circle: a radius or a diameter?

Example 7

easy

A circle has radius

3

. Write its circumference in terms of

\pi

Radius = 3 — write the circumference in terms of π

Example 8

easy

A circle has radius

4

. Write its area in terms of

\pi

Radius = 4 — write the area in terms of π

Example 9

easy

A straight line segment from one point on a circle to another is called a what?

Example 10

easy

Using

\pi \approx 3.14

, estimate the circumference of a circle with diameter

10

Diameter = 10 — estimate the circumference using π ≈ 3.14

Example 11

medium

A circle has circumference

20\pi

. Find its radius.

Circumference = 20π — find the radius

Example 12

medium

A circle has area

49\pi

. Find its radius.

Area = 49π — find the radius

Example 13

medium

If you double a circle's radius, what happens to its circumference?

Example 14

medium

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

Example 15

medium

A square has side

10

. A circle is inscribed inside it, touching all four sides. Find the circle's radius.

Example 16

medium

A bicycle wheel has radius

30

cm. About how far does it travel in one full turn? Use

\pi \approx 3.14

Example 17

medium

Two circles have radii

3

and

6

. What is the ratio of their areas?

Example 18

medium

A semicircle has radius

6

. Find its perimeter in terms of

\pi

Semicircle with radius 6 — find the perimeter

Example 19

challenge

A circle is inscribed in a square of side

8

, and another circle is circumscribed about the same square. Find the ratio of the larger circle's area to the smaller circle's area.

Example 20

challenge

Four identical coins of radius

1

are packed snugly in a square so each touches two neighbors. Find the area of the gap in the center between the four coins.

Example 21

challenge

A goat is tied by a

7

-meter rope to a corner on the outside of a square barn with side

10

meters. Over what area can the goat graze?

Example 22

challenge

Explain why, if a circle's circumference equals its area numerically, the radius must be

2

Example 23

easy

A circle has radius

9

. What is its diameter?

Radius = 9 — find the diameter

Example 24

easy

A circle has diameter

26

. What is its radius?

Diameter = 26 — find the radius

Example 25

easy

A circle has radius

6

. Find its area in terms of

\pi

Radius = 6 — find the area in terms of π

Example 26

easy

A circle has radius

8

. Find its circumference in terms of

\pi

Radius = 8 — find the circumference in terms of π

Example 27

easy

A circular pizza has radius

7

inches. Use

\pi\approx 3.14

to estimate its circumference.

Example 28

medium

A circle has area

25\pi

. Find its radius.

Area = 25π — find the radius

Example 29

medium

A circle has circumference

30\pi

. Find its diameter.

Example 30

medium

A semicircle has diameter

20

. Find its perimeter in terms of

\pi

Semicircle with diameter 20 — find the perimeter

Example 31

medium

Two circles have radii

4

and

10

. Find the ratio of their circumferences.

Example 32

medium

Two circles have radii

3

and

9

. Find the ratio of their areas.

Example 33

medium

A square has side

10

. Find the radius of the circumscribed circle (passing through all

4

corners).

Example 34

hard

A circle has area

A

equal to circumference

C

numerically. Find

r

Example 35

hard

Three circles of radius

1

are arranged so each touches the other two. Find the side length of the equilateral triangle joining centers.

Example 36

hard

A circle is inscribed in an equilateral triangle of side

6

. Find its radius.

Example 37

hard

A circle has center

(2,3)

and passes through

(5,7)

. Find its radius.

Example 38

hard

A circle has equation

(x-3)^2+(y+2)^2=16

. State its center and radius.

Example 39

challenge

A circle of radius

5

has a chord of length

6

. Find the distance from the center.

Radius = 5, chord = 6 — find the distance from the center

Example 40

challenge

A circle has area

A

that is twice its circumference

C

numerically. Find

r

Example 41

challenge

A circle is inscribed in a regular hexagon of side

4

. Find the circle's radius.

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Related Concepts

Basic Shapes Circumference Circles Pi (π)

Background Knowledge

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

shapes