Equation 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 Equation 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 standard form equation $(x - h)^2 + (y - k)^2 = r^2$ describes a circle with center $(h, k)$ and radius $r$ in the coordinate plane.

A circle is the set of all points at the same distance (the radius) from a center point. The equation just says 'the distance from $(x, y)$ to the center $(h, k)$ equals $r$ ,' using the distance formula squared.

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: $(x-h)^2+(y-k)^2=r^2$ collects every point exactly $r$ from $(h,k)$ .

Common stuck point: The procedure for equation of a circle is the easy part; the trap is reading the center sign backwards. Asking "Are $x^2$ and $y^2$ present with equal positive coefficients and a constant on the other side?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are $x^2$ and $y^2$ present with equal positive coefficients and a constant on the other side?

Worked Examples

Example 1

easy

Write the equation of the circle with center

(3, -2)

and radius

5

Answer

(x - 3)^2 + (y + 2)^2 = 25

First step

The standard form of a circle's equation is

(x - h)^2 + (y - k)^2 = r^2

, where

(h, k)

is the center and

r

is the radius.

Full solution

2
Substitute $h = 3$ , $k = -2$ , $r = 5$ .
3
$(x - 3)^2 + (y + 2)^2 = 25$ .

The equation of a circle is derived from the distance formula: every point

(x, y)

on the circle is exactly

r

units from the center

(h, k)

. This gives

\sqrt{(x-h)^2 + (y-k)^2} = r

, which when squared yields the standard form.

Example 2

medium

Find the center and radius of the circle

x^2 + y^2 - 6x + 4y - 12 = 0

Example 3

medium

Convert

x^2 + y^2 - 4x + 6y - 3 = 0

to standard form and identify the center and radius.

Example 4

medium

A diameter of a circle has endpoints

(2, 3)

and

(8, 11)

. Find the equation.

Example 5

medium

Determine whether

x^2 + y^2 - 2x + 4y + 10 = 0

describes a real circle.

Example 6

hard

Find the equation of the circle passing through

(1, 0)

(5, 0)

, and

(3, 4)

Example 7

hard

Find the equation of the circle tangent to both the

x

- and

y

-axes in the first quadrant with radius

r = 6

Example 8

hard

Determine the relationship (intersecting, tangent, or disjoint) between the circles

(x - 1)^2 + (y - 1)^2 = 4

and

(x - 5)^2 + (y - 4)^2 = 9

Example 9

challenge

Find all values of

b

for which the line

y = x + b

is tangent to the circle

x^2 + y^2 = 8

Practice Problems

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

Example 1

medium

Find the equation of the circle that has a diameter with endpoints

(-1, 4)

and

(5, -2)

Example 2

hard

Determine whether the circles

(x-1)^2 + (y-3)^2 = 16

and

(x-7)^2 + (y-3)^2 = 4

intersect, and find the number of intersection points.

Example 3

easy

Give the center and radius of

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

Example 4

easy

Give the center of

(x+5)^2 + (y-1)^2 = 9

Example 5

easy

Write the equation of a circle with center

(0,0)

and radius 7.

Example 6

easy

Write the equation of a circle with center

(2,-3)

and radius 5.

Example 7

easy

What is the radius of

x^2 + y^2 = 36

Example 8

easy

Does the point

(3,4)

lie on

x^2+y^2=25

Example 9

easy

Write the equation of a circle centered at

(1,1)

with radius

\sqrt{10}

Example 10

easy

Give the radius of

(x-4)^2 + (y+2)^2 = 49

Example 11

medium

Find the center and radius of

x^2 + y^2 - 6x + 4y - 12 = 0

Example 12

medium

Find the center and radius of

x^2 + y^2 + 8x - 2y + 8 = 0

Example 13

medium

Find the equation of the circle with center

(2,3)

passing through

(5,7)

Example 14

medium

Find the equation of the circle with endpoints of a diameter at

(1,2)

and

(7,10)

Example 15

medium

x^2 + y^2 + 2x + 2y + 2 = 0

a circle? If so, give center and radius.

Example 16

medium

Find

r

(x-1)^2 + (y+4)^2 = k

passes through

(4,0)

Example 17

medium

Where does

x^2 + y^2 = 25

intersect the line

x = 3

Example 18

challenge

Find the equation of the circle through

(0,0)

(6,0)

, and

(0,8)

Example 19

challenge

For what values of

k

does

x^2+y^2-4x+2y+k=0

represent a real circle?

Example 20

challenge

A circle is tangent to the

x

-axis at

(4,0)

with radius 3. Find its equation(s).

Example 21

medium

Find the center and radius of

2x^2 + 2y^2 - 8x + 12y - 6 = 0

Example 22

medium

Find where

x^2+y^2=20

meets the line

y=2x

Example 23

easy

Find the center and radius of

(x + 1)^2 + (y - 4)^2 = 25

Example 24

easy

Write the equation of a circle with center

(-3, 0)

and radius

4

Example 25

easy

Does

(2, -1)

lie on the circle

(x - 2)^2 + (y + 1)^2 = 0.0001

Example 26

easy

Write the equation of the circle centered at

(4, -5)

tangent to the

x

-axis.

Example 27

medium

Find the equation of the circle centered at

(1, 2)

passing through

(4, 6)

Example 28

medium

Find the center and radius of

x^2 + y^2 + 8x - 2y + 1 = 0

Example 29

medium

Find the equation of the circle centered at the origin and passing through

(5, 12)

Example 30

medium

Find

k

so that

x^2 + y^2 - 6x + 4y + k = 0

represents a circle of radius

5

Example 31

medium

Write the equation of a circle centered at

(2, -1)

tangent to the line

y = 3

Example 32

hard

Find the points where the circle

x^2 + y^2 = 25

intersects the line

y = x + 1

Example 33

hard

Find the length of the chord cut by the circle

x^2 + y^2 = 25

on the line

y = 3

Example 34

hard

Find the area of the region inside the circle

x^2 + y^2 - 6x - 8y = 0

← Back to Equation of a Circle Formula Explained Practice Mode

Related Concepts

Pythagorean Theorem Domain Ellipse Hyperbola Conic Sections Overview

Background Knowledge

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

pythagorean theoremdomain