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Polynomials & Functions

Equations of Circles

The standard equation of a circle is (x − h)² + (y − k)² = r², where (h, k) is the center and r is the radius.

✨ The equation of a circle tells you exactly where it lives and how big it is — two facts, one equation!

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Worked Examples

Follow along with these step-by-step examples. Take your time — there's no rush!

1Example 1

Problem

Write the equation of a circle with center (3, −2) and radius 5. Then look at the graph.

xy-10-10-8-8-6-6-4-4-2-22244668810100r = 5(3, -2)

Step-by-Step Solution

1

Use (x − h)² + (y − k)² = r²

2

h = 3, k = −2, r = 5

3

(x − 3)² + (y − (−2))² = 25

4

(x − 3)² + (y + 2)² = 25

Answer

(x − 3)² + (y + 2)² = 25

2Example 2

Problem

Find the center and radius of (x + 1)² + (y − 4)² = 36. The graph shows the circle.

xy-12-12-10-10-8-8-6-6-4-4-2-222446688101012120r = 6(-1, 4)

Step-by-Step Solution

1

Rewrite: (x − (−1))² + (y − 4)² = 36

2

Center: (−1, 4)

3

r² = 36, so r = 6

Answer

Center (−1, 4), radius = 6

3Example 3

Problem

Write the equation of a circle centered at the origin with radius 7.

xy-9-9-7-7-5-5-3-3-1-111335577990r = 7(0, 0)

Step-by-Step Solution

1

Center (0, 0), r = 7.

2

(x − 0)² + (y − 0)² = 49

3

x² + y² = 49

Answer

x² + y² = 49

4Example 4

Problem

A circle has equation (x − 5)² + (y + 1)² = 16. What is the radius? See the graph.

xy-11-11-9-9-7-7-5-5-3-3-1-1113355779911110r = 4(5, -1)

Step-by-Step Solution

1

r² = 16

2

r = √16 = 4

Answer

Radius = 4

5Example 5

Problem

Does the point (3, 4) lie on the circle x² + y² = 25? The graph shows the circle and the point.

xy-7-7-6-6-5-5-4-4-3-3-2-2-1-1112233445566770r = 5(0, 0)(3, 4)

Step-by-Step Solution

1

Substitute: (3)² + (4)² = 9 + 16 = 25.

2

25 = 25 ✓ — the point satisfies the equation.

3

Yes, the point lies on the circle.

Answer

Yes — (3)² + (4)² = 25 ✓

Your Turn — Practice Problems

Try all 5 problems on your own first. Write out your work — that's how it sticks!

💡 Tip: Don't peek at the answers until you've genuinely tried each one.

1

Write the equation. Center (0, 0), radius 4.

xy-6-6-5-5-4-4-3-3-2-2-1-11122334455660r = 4(0, 0)
2

Write the equation. Center (2, 5), radius 3.

xy-10-10-8-8-6-6-4-4-2-22244668810100r = 3(2, 5)
3

Find the center and radius from the graph.

xy-11-11-9-9-7-7-5-5-3-3-1-1113355779911110r = 5(4, -3)
4

Find the center and radius. Equation: x² + y² = 100.

xy-12-12-10-10-8-8-6-6-4-4-2-222446688101012120r = 10(0, 0)
5

Write the equation. Center (−3, 1), radius 8.

xy-13-13-11-11-9-9-7-7-5-5-3-3-1-11133557799111113130r = 8(-3, 1)
6

Does (0, 5) lie on the circle x² + y² = 25? Check the graph.

xy-7-7-6-6-5-5-4-4-3-3-2-2-1-1112233445566770r = 5(0, 0)(0, 5)
7

Find the center and radius from the graph.

xy-16-16-14-14-12-12-10-10-8-8-6-6-4-4-2-22244668810101212141416160r = 7(-2, 7)
8

Write the equation. Center (1, −4), radius 6.

xy-12-12-10-10-8-8-6-6-4-4-2-222446688101012120r = 6(1, -4)
9

Does (3, 4) lie on x² + y² = 16? Check the graph.

xy-6-6-5-5-4-4-3-3-2-2-1-11122334455660r = 4(0, 0)(3, 4)
10

Find the radius of x² + y² = 144.

xy-14-14-12-12-10-10-8-8-6-6-4-4-2-2224466881010121214140r = 12(0, 0)
11

Write the equation. Center (0, 3), radius 5.

xy-10-10-8-8-6-6-4-4-2-22244668810100r = 5(0, 3)
12

Find the center and radius from the graph.

xy-17-17-15-15-13-13-11-11-9-9-7-7-5-5-3-3-1-1113355779911111313151517170r = 9(6, 0)

Finished all 12? Give yourself a pat on the back — then check your work!

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