Worksheets · Precalculus

Equations of circles worksheet for Precalculus

Precalculus treats the circle as the first conic section: write its equation from a center and a point, a diameter or an area, or start from general form and complete the square to find the center and radius. Terms can be spread over both sides, and the same routine comes back for ellipses and hyperbolas.

Also taught in Algebra 2 and Geometry, each at its own level and with its own examples.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Equations of Circles

Date Period

Answer each question.

  1. Write the equation from the center and radius or a point

    Write the equation of the circle with center (7,−5) passing through (21,−5)
  2. Write the equation of the circle with center (1,−5) that passes through (0,−3)
  3. Write the equation of the circle with center (−4,−13) passing through (11,−13)
  4. Write the equation of the circle with center (−6,4) that passes through (−12,−4)
  5. Write the equation of the circle with center (10,10) passing through (25,10)
  6. Write the equation of the circle with center (3,6) that passes through (9,14)
  7. Write the equation of the circle with center (−13,11) passing through (1,11)

  1. Find the center and radius

    Find the center and radius of x2+y2−2x−18y−87=0
  2. Find the center and radius of x2+14x+1=−y2−22y.
  3. Find the center and radius of x2+y2+20x+22y+25=0
  4. Find the center and radius of x2+y2=22x+16y−104.
  5. Find the center and radius of x2+y2+10x−20y−71=0
  6. Find the center and radius of x2−12x+20=−y2+6y.
  7. Find the center and radius of x2+y2+14y−72=0

  1. Write the equation from a diameter

    Write the equation of the circle with a diameter from (−2,2) to (2,−6)
  2. Write the equation of the circle with a diameter from (4,−10) to (6,−6)
  3. Write the equation of the circle with a diameter from (9,5) to (−11,−11)

  1. Write the equation from the area, circumference or a tangent line

    Write the equation of the circle with center (12,−12) that is tangent to the line y=−10
  2. Write the equation of the circle with center (−1,11) and area 121π
  3. Write the equation of the circle with center (−3,9) and circumference 28π

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Equations of CirclesAnswer keyVersion 2

Date Period

Answer each question.

  1. Write the equation from the center and radius or a point

    Write the equation of the circle with center (7,−5) passing through (21,−5)(x−7)2+(y+5)2=196
  2. Write the equation of the circle with center (1,−5) that passes through (0,−3)(x−1)2+(y+5)2=5
  3. Write the equation of the circle with center (−4,−13) passing through (11,−13)(x+4)2+(y+13)2=225
  4. Write the equation of the circle with center (−6,4) that passes through (−12,−4)(x+6)2+(y−4)2=100
  5. Write the equation of the circle with center (10,10) passing through (25,10)(x−10)2+(y−10)2=225
  6. Write the equation of the circle with center (3,6) that passes through (9,14)(x−3)2+(y−6)2=100
  7. Write the equation of the circle with center (−13,11) passing through (1,11)(x+13)2+(y−11)2=196

  1. Find the center and radius

    Find the center and radius of x2+y2−2x−18y−87=0Center (1,9), radius 13
  2. Find the center and radius of x2+14x+1=−y2−22y.Center (−7,−11), radius 13
  3. Find the center and radius of x2+y2+20x+22y+25=0Center (−10,−11), radius 14
  4. Find the center and radius of x2+y2=22x+16y−104.Center (11,8), radius 9
  5. Find the center and radius of x2+y2+10x−20y−71=0Center (−5,10), radius 14
  6. Find the center and radius of x2−12x+20=−y2+6y.Center (6,3), radius 5
  7. Find the center and radius of x2+y2+14y−72=0Center (0,−7), radius 11

  1. Write the equation from a diameter

    Write the equation of the circle with a diameter from (−2,2) to (2,−6)x2+(y+2)2=20
  2. Write the equation of the circle with a diameter from (4,−10) to (6,−6)(x−5)2+(y+8)2=5
  3. Write the equation of the circle with a diameter from (9,5) to (−11,−11)(x+1)2+(y+3)2=164

  1. Write the equation from the area, circumference or a tangent line

    Write the equation of the circle with center (12,−12) that is tangent to the line y=−10(x−12)2+(y+12)2=4
  2. Write the equation of the circle with center (−1,11) and area 121π(x+1)2+(y−11)2=121
  3. Write the equation of the circle with center (−3,9) and circumference 28π(x+3)2+(y−9)2=196

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Find the center and radius:
x2+y2−4x+6y−12=0

  1. Group and move the constant: x² - 4x + y² + 6y = 12.
  2. Complete the square, adding 4 and 9 to both sides: (x - 2)² + (y + 3)² = 25.

The answer is Center (2,−3), radius 5.

Where students go wrong

Adding the completing-the-square numbers to one side only. The 4 and the 9 go on both sides, so the right side is 12 + 4 + 9 = 25 and the radius is 5, not √12.

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 → x2+(y−1)2=49
Find the center and radius from the general form
Find the center and radius of x2+y2−2y−35=0 → Center (0,1), radius 6
From the ends of a diameter
Write the equation of the circle with a diameter from (0,1) to (2,1) → (x−1)2+(y−1)2=1
From the center and a point
Write the equation of the circle with center (1,3) that passes through (1,8) → (x−1)2+(y−3)2=25
From the center and the area, circumference or a tangent line
Write the equation of the circle with center (0,1) and circumference 4π → x2+(y−1)2=4
General form spread over both sides
Find the center and radius of x2+y2=−6x+10y−18. → Center (−3,5), radius 4
A radius that is a root
Find the center and radius: (x−6)2+(y−1)2=12 → Center (6,1), radius 23
Tangent to three lines
A circle is tangent to the lines x=−3, x=3 and y=3, and lies above y=3. Write its equation. → x2+(y−6)2=9
Graph a circle
Graph (x−1)2+(y−3)2=9 → Center (1,3), radius 3
More circle problems
Write the equation of the circle whose center is on the y-axis and that is tangent to y=6 and y=−14 → x2+(y+4)2=100
Through three points, general form
Write, in general form, the circle through (3,3), (−3,3), (−2,4) → x2+y2−2y−12=0
A square-root radius
Graph x2+(y−1)2=15 → Center (0,1), radius 15≈3.9

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 12: write the equation from center and radius, find the center and radius from the general form, from the ends of a diameter, from the center and a point, from the center and the area, circumference or a tangent line, general form spread over both sides, a radius that is a root, tangent to three lines, graph a circle, more circle problems, through three points, general form and a square-root radius. 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.