Precalculus equations of circles worksheet

The equation of a circle says one thing: the distance from the center is the radius. Going the other way means completing the square twice, once in x and once in y.

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Sigma Prep · sigmaprep.io/worksheets Name Equations of Circles Date Period Answer each question. Write the equation of the circle with center $(4, - 14)$ passing through $(19, - 14)$ $x^{2} + y^{2} - 24 x - 24 y + 144 = 0$ Write the equation of the circle with center $(3, 6)$ passing through $(17, 6)$ $x^{2} + y^{2} - 24 x + 2 y + 1 = 0$ Write the equation of the circle with center $(- 10, - 3)$ passing through $(3, - 3)$ $x^{2} + y^{2} + 18 x - 40 = 0$ Write the equation of the circle with center $(- 3, 8)$ passing through $(11, 8)$ $x^{2} + y^{2} + 16 x + 22 y + 41 = 0$ Write the equation of the circle with center $(3, 9)$ passing through $(17, 9)$ $x^{2} + y^{2} + 14 x - 8 y - 104 = 0$ Write the equation of the circle with center $(7, - 9)$ passing through $(22, - 9)$ $x^{2} + y^{2} + 2 x - 14 y - 146 = 0$ Write the equation of the circle with center $(- 8, 7)$ passing through $(5, 7)$ $x^{2} + y^{2} + 8 x + 4 y - 101 = 0$ Write the equation of the circle with center $(- 10, 7)$ passing through $(5, 7)$ $x^{2} + y^{2} + 18 x + 12 y - 4 = 0$ Write the equation of the circle with center $(4, 11)$ passing through $(17, 11)$ $x^{2} + y^{2} - 24 x - 10 y + 25 = 0$ Write the equation of the circle with center $(- 6, 5)$ passing through $(10, 5)$ $x^{2} + y^{2} - 22 x - 10 y - 50 = 0$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Equations of Circles

Date Period

Answer each question.

Write the equation of the circle with center $(4, - 14)$ passing through $(19, - 14)$
$x^{2} + y^{2} - 24 x - 24 y + 144 = 0$
Write the equation of the circle with center $(3, 6)$ passing through $(17, 6)$
$x^{2} + y^{2} - 24 x + 2 y + 1 = 0$
Write the equation of the circle with center $(- 10, - 3)$ passing through $(3, - 3)$
$x^{2} + y^{2} + 18 x - 40 = 0$
Write the equation of the circle with center $(- 3, 8)$ passing through $(11, 8)$
$x^{2} + y^{2} + 16 x + 22 y + 41 = 0$
Write the equation of the circle with center $(3, 9)$ passing through $(17, 9)$
$x^{2} + y^{2} + 14 x - 8 y - 104 = 0$
Write the equation of the circle with center $(7, - 9)$ passing through $(22, - 9)$
$x^{2} + y^{2} + 2 x - 14 y - 146 = 0$
Write the equation of the circle with center $(- 8, 7)$ passing through $(5, 7)$
$x^{2} + y^{2} + 8 x + 4 y - 101 = 0$
Write the equation of the circle with center $(- 10, 7)$ passing through $(5, 7)$
$x^{2} + y^{2} + 18 x + 12 y - 4 = 0$
Write the equation of the circle with center $(4, 11)$ passing through $(17, 11)$
$x^{2} + y^{2} - 24 x - 10 y + 25 = 0$
Write the equation of the circle with center $(- 6, 5)$ passing through $(10, 5)$
$x^{2} + y^{2} - 22 x - 10 y - 50 = 0$



Sigma Prep · sigmaprep.io/worksheets Name Equations of CirclesAnswer keyVersion 1 Date Period Answer each question. Write the equation of the circle with center $(4, - 14)$ passing through $(19, - 14)$ ${(x - 4)}^{2} + {(y + 14)}^{2} = 225$ $x^{2} + y^{2} - 24 x - 24 y + 144 = 0$ Center $(12, 12)$ , radius $12$ Write the equation of the circle with center $(3, 6)$ passing through $(17, 6)$ ${(x - 3)}^{2} + {(y - 6)}^{2} = 196$ $x^{2} + y^{2} - 24 x + 2 y + 1 = 0$ Center $(12, - 1)$ , radius $12$ Write the equation of the circle with center $(- 10, - 3)$ passing through $(3, - 3)$ ${(x + 10)}^{2} + {(y + 3)}^{2} = 169$ $x^{2} + y^{2} + 18 x - 40 = 0$ Center $(- 9, 0)$ , radius $11$ Write the equation of the circle with center $(- 3, 8)$ passing through $(11, 8)$ ${(x + 3)}^{2} + {(y - 8)}^{2} = 196$ $x^{2} + y^{2} + 16 x + 22 y + 41 = 0$ Center $(- 8, - 11)$ , radius $12$ Write the equation of the circle with center $(3, 9)$ passing through $(17, 9)$ ${(x - 3)}^{2} + {(y - 9)}^{2} = 196$ $x^{2} + y^{2} + 14 x - 8 y - 104 = 0$ Center $(- 7, 4)$ , radius $13$ Write the equation of the circle with center $(7, - 9)$ passing through $(22, - 9)$ ${(x - 7)}^{2} + {(y + 9)}^{2} = 225$ $x^{2} + y^{2} + 2 x - 14 y - 146 = 0$ Center $(- 1, 7)$ , radius $14$ Write the equation of the circle with center $(- 8, 7)$ passing through $(5, 7)$ ${(x + 8)}^{2} + {(y - 7)}^{2} = 169$ $x^{2} + y^{2} + 8 x + 4 y - 101 = 0$ Center $(- 4, - 2)$ , radius $11$ Write the equation of the circle with center $(- 10, 7)$ passing through $(5, 7)$ ${(x + 10)}^{2} + {(y - 7)}^{2} = 225$ $x^{2} + y^{2} + 18 x + 12 y - 4 = 0$ Center $(- 9, - 6)$ , radius $11$ Write the equation of the circle with center $(4, 11)$ passing through $(17, 11)$ ${(x - 4)}^{2} + {(y - 11)}^{2} = 169$ $x^{2} + y^{2} - 24 x - 10 y + 25 = 0$ Center $(12, 5)$ , radius $12$ Write the equation of the circle with center $(- 6, 5)$ passing through $(10, 5)$ ${(x + 6)}^{2} + {(y - 5)}^{2} = 256$ $x^{2} + y^{2} - 22 x - 10 y - 50 = 0$ Center $(11, 5)$ , radius $14$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Equations of CirclesAnswer keyVersion 1

Date Period

Answer each question.

Write the equation of the circle with center $(4, - 14)$ passing through $(19, - 14)$ ${(x - 4)}^{2} + {(y + 14)}^{2} = 225$
$x^{2} + y^{2} - 24 x - 24 y + 144 = 0$ Center $(12, 12)$ , radius $12$
Write the equation of the circle with center $(3, 6)$ passing through $(17, 6)$ ${(x - 3)}^{2} + {(y - 6)}^{2} = 196$
$x^{2} + y^{2} - 24 x + 2 y + 1 = 0$ Center $(12, - 1)$ , radius $12$
Write the equation of the circle with center $(- 10, - 3)$ passing through $(3, - 3)$ ${(x + 10)}^{2} + {(y + 3)}^{2} = 169$
$x^{2} + y^{2} + 18 x - 40 = 0$ Center $(- 9, 0)$ , radius $11$
Write the equation of the circle with center $(- 3, 8)$ passing through $(11, 8)$ ${(x + 3)}^{2} + {(y - 8)}^{2} = 196$
$x^{2} + y^{2} + 16 x + 22 y + 41 = 0$ Center $(- 8, - 11)$ , radius $12$
Write the equation of the circle with center $(3, 9)$ passing through $(17, 9)$ ${(x - 3)}^{2} + {(y - 9)}^{2} = 196$
$x^{2} + y^{2} + 14 x - 8 y - 104 = 0$ Center $(- 7, 4)$ , radius $13$
Write the equation of the circle with center $(7, - 9)$ passing through $(22, - 9)$ ${(x - 7)}^{2} + {(y + 9)}^{2} = 225$
$x^{2} + y^{2} + 2 x - 14 y - 146 = 0$ Center $(- 1, 7)$ , radius $14$
Write the equation of the circle with center $(- 8, 7)$ passing through $(5, 7)$ ${(x + 8)}^{2} + {(y - 7)}^{2} = 169$
$x^{2} + y^{2} + 8 x + 4 y - 101 = 0$ Center $(- 4, - 2)$ , radius $11$
Write the equation of the circle with center $(- 10, 7)$ passing through $(5, 7)$ ${(x + 10)}^{2} + {(y - 7)}^{2} = 225$
$x^{2} + y^{2} + 18 x + 12 y - 4 = 0$ Center $(- 9, - 6)$ , radius $11$
Write the equation of the circle with center $(4, 11)$ passing through $(17, 11)$ ${(x - 4)}^{2} + {(y - 11)}^{2} = 169$
$x^{2} + y^{2} - 24 x - 10 y + 25 = 0$ Center $(12, 5)$ , radius $12$
Write the equation of the circle with center $(- 6, 5)$ passing through $(10, 5)$ ${(x + 6)}^{2} + {(y - 5)}^{2} = 256$
$x^{2} + y^{2} - 22 x - 10 y - 50 = 0$ Center $(11, 5)$ , radius $14$

How to do these

Write the equation of the circle with center $(3, - 2)$ and radius $5$

The center goes in with the opposite sign.
The radius is squared on the right.

The answer is ${(x - 3)}^{2} + {(y + 2)}^{2} = 25$ .

Where students go wrong

Copying the center in with its own sign. A center of -2 gives (y + 2), because the equation measures the distance from it. The other slip is writing r rather than r squared on the right.

Also called standard form of a circle, completing the square for circles or center and radius.

What you can put on this worksheet

Write the equation from center and radius: Write the equation of the circle with center $(0, 1)$ and radius $7$ → ${(x)}^{2} + {(y - 1)}^{2} = 49$
Find the center and radius from the general form: $x^{2} + y^{2} - 2 y - 35 = 0$ → Center $(0, 1)$ , radius $6$

Questions about these worksheets

Yes. Every equations of circles sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the equations of circles questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 2: write the equation from center and radius and find the center and radius from the general form. Tick as many as you want and set how many of each, or let it spread them evenly.

Copying the center in with its own sign. A center of -2 gives (y + 2), because the equation measures the distance from it. The other slip is writing r rather than r squared on the right.

Precalculus equations of circles worksheet

Select your difficulty

Equations of Circles

Equations of CirclesAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

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