Worksheets · Precalculus

Polar and rectangular equations worksheet

Both forms describe the same curve. Going one way you replace x with r cos and y with r sin; going the other you look for the pieces of x squared plus y squared and put them back together.

Select your difficulty

Pick more than one for a sheet that mixes them.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Polar and Rectangular Forms of Equations

Date Period

Write each equation in the other form.

  1. Write in polar form: x2+y2=81
  2. Write in rectangular form: tan⁡θ=−5
  3. Write in polar form: x2+y2=−9y
  4. Write in rectangular form: r=−10csc⁡θ
  5. Write in polar form: y=−8
  6. Write in rectangular form: r=−8csc⁡θ
  7. Write in polar form: x2+y2=64
  8. Write in rectangular form: r=9cos⁡θ
  9. Write in polar form: x=8
  10. Write in rectangular form: r=−9sin⁡θ
  11. Write in polar form: y=10
  12. Write in rectangular form: r=10sec⁡θ
  13. Write in polar form: x=−8
  14. Write in rectangular form: r=8
  15. Write in polar form: x2+y2=−10x
  16. Write in rectangular form: tan⁡θ=2
  17. Write in polar form: y=−10
  18. Write in rectangular form: tan⁡θ=5
  19. Write in polar form: y=−5x
  20. Write in rectangular form: tan⁡θ=3

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

Sigma Prep · sigmaprep.io/worksheets

Name

Polar and Rectangular Forms of EquationsAnswer keyVersion 1

Date Period

Write each equation in the other form.

  1. Write in polar form: x2+y2=81r=9
  2. Write in rectangular form: tan⁡θ=−5y=−5x
  3. Write in polar form: x2+y2=−9yr=−9sin⁡θ
  4. Write in rectangular form: r=−10csc⁡θy=−10
  5. Write in polar form: y=−8r=−8csc⁡θ
  6. Write in rectangular form: r=−8csc⁡θy=−8
  7. Write in polar form: x2+y2=64r=8
  8. Write in rectangular form: r=9cos⁡θx2+y2=9x
  9. Write in polar form: x=8r=8sec⁡θ
  10. Write in rectangular form: r=−9sin⁡θx2+y2=−9y
  11. Write in polar form: y=10r=10csc⁡θ
  12. Write in rectangular form: r=10sec⁡θx=10
  13. Write in polar form: x=−8r=−8sec⁡θ
  14. Write in rectangular form: r=8x2+y2=64
  15. Write in polar form: x2+y2=−10xr=−10cos⁡θ
  16. Write in rectangular form: tan⁡θ=2y=2x
  17. Write in polar form: y=−10r=−10csc⁡θ
  18. Write in rectangular form: tan⁡θ=5y=5x
  19. Write in polar form: y=−5xtan⁡θ=−5
  20. Write in rectangular form: tan⁡θ=3y=3x

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

How to do these

Write in polar form: x2+y2=4x

  1. The left side is r squared.
  2. The right side is 4 r cos theta.
  3. Divide both sides by r.

The answer is r=4cos⁡θ.

Where students go wrong

Forgetting that dividing by r is allowed here. r squared equals 4 r cos theta looks stuck until you notice r is a common factor, and the pole is still on the curve afterwards, so nothing is lost.

Also called converting polar equations, polar form of an equation or rectangular form.

What you can put on this worksheet

Rectangular equation to polar
Write in polar form: x=5 → r=5sec⁡θ
Polar equation to rectangular
Write in rectangular form: r=5sec⁡θ → x=5

Questions about these worksheets

Yes. Every polar and rectangular equations 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 polar and rectangular equations 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: rectangular equation to polar and polar equation to rectangular. Tick as many as you want and set how many of each, or let it spread them evenly.

Forgetting that dividing by r is allowed here. r squared equals 4 r cos theta looks stuck until you notice r is a common factor, and the pole is still on the curve afterwards, so nothing is lost.