Worksheets · Calculus

Arc length of parametric and polar curves worksheet

For a parametric curve the length element is √((dx/dt)² + (dy/dt)²) dt; for a polar curve it is √(r² + (dr/dθ)²) dθ. Each curve here is one where that root simplifies, so the integral can be done by hand.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Arc Length: Parametric and Polar

Date Period

Find the length of each curve. Give exact answers.

  1. Find the length of the curve x=t22, y=13(2t+1)3/2, for 0≤t≤3.
  2. Find the length of the polar curve r=e2θ, for 0≤θ≤ln⁡9.
  3. Find the length of the curve x=6t2, y=4t3, for 34≤t≤43.
  4. Find the length of the polar curve r=θ2, for 0≤θ≤5.
  5. Find the length of the curve x=3t2, y=2t3, for 43≤t≤22.
  6. Find the length of the polar curve r=2eθ, for 0≤θ≤ln⁡3.
  7. Find the length of the curve x=12t2, y=8t3, for 0≤t≤22.
  8. Find the length of the polar curve r=eθ, for 0≤θ≤ln⁡4.
  9. Find the length of the curve x=9t2, y=6t3, for 43≤t≤3.
  10. Find the length of the polar curve r=3eθ, for 0≤θ≤ln⁡7.
  11. Find the length of the curve x=9t2, y=6t3, for 0≤t≤3.
  12. Find the length of the polar curve r=θ2, for 0≤θ≤23.
  13. Find the length of the curve x=6t2, y=4t3, for 3≤t≤15.
  14. Find the length of the polar curve r=e2θ, for 0≤θ≤ln⁡6.
  15. Find the length of the curve x=3t2, y=2t3, for 43≤t≤3.
  16. Find the length of the polar curve r=3eθ, for 0≤θ≤ln⁡8.
  17. Find the length of the curve x=12t2, y=8t3, for 3≤t≤22.
  18. Find the length of the polar curve r=2eθ, for 0≤θ≤ln⁡4.
  19. Find the length of the curve x=t22, y=13(2t+1)3/2, for 0≤t≤2.
  20. Find the length of the polar curve r=eθ, for 0≤θ≤ln⁡6.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Arc Length: Parametric and PolarAnswer keyVersion 1

Date Period

Find the length of each curve. Give exact answers.

  1. Find the length of the curve x=t22, y=13(2t+1)3/2, for 0≤t≤3.152
  2. Find the length of the polar curve r=e2θ, for 0≤θ≤ln⁡9.405
  3. Find the length of the curve x=6t2, y=4t3, for 34≤t≤43.4625432
  4. Find the length of the polar curve r=θ2, for 0≤θ≤5.193
  5. Find the length of the curve x=3t2, y=2t3, for 43≤t≤22.120827
  6. Find the length of the polar curve r=2eθ, for 0≤θ≤ln⁡3.42
  7. Find the length of the curve x=12t2, y=8t3, for 0≤t≤22.208
  8. Find the length of the polar curve r=eθ, for 0≤θ≤ln⁡4.32
  9. Find the length of the curve x=9t2, y=6t3, for 43≤t≤3.1829
  10. Find the length of the polar curve r=3eθ, for 0≤θ≤ln⁡7.182
  11. Find the length of the curve x=9t2, y=6t3, for 0≤t≤3.42
  12. Find the length of the polar curve r=θ2, for 0≤θ≤23.563
  13. Find the length of the curve x=6t2, y=4t3, for 3≤t≤15.224
  14. Find the length of the polar curve r=e2θ, for 0≤θ≤ln⁡6.3552
  15. Find the length of the curve x=3t2, y=2t3, for 43≤t≤3.18227
  16. Find the length of the polar curve r=3eθ, for 0≤θ≤ln⁡8.212
  17. Find the length of the curve x=12t2, y=8t3, for 3≤t≤22.152
  18. Find the length of the polar curve r=2eθ, for 0≤θ≤ln⁡4.62
  19. Find the length of the curve x=t22, y=13(2t+1)3/2, for 0≤t≤2.4
  20. Find the length of the polar curve r=eθ, for 0≤θ≤ln⁡6.52

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

How to do these

Find the length of the curve x=3t2, y=2t3, 0≤t≤3.

  1. dx/dt = 6t and dy/dt = 6t², so the speed is √(36t² + 36t⁴) = 6t√(1 + t²).
  2. L = ∫ from 0 to √3 of 6t√(1 + t²) dt = 2[(1 + t²)^(3/2)] from 0 to √3.
  3. 2(4^(3/2) - 1) = 2(8 - 1) = 14.

The answer is 14.

Where students go wrong

Leaving out r² in the polar formula and integrating |dr/dθ| alone. A circle r = 3 has dr/dθ = 0 and still has length: the r² term is the part of the motion that goes around.

Also called parametric arc length, polar arc length, length of a parametric curve or length of a polar curve.

What you can put on this worksheet

Parametric curves
Find the length of the curve x=12t, y=−5t+1, for 0≤t≤4. → 52
Polar curves
Find the length of the polar curve r=8cos⁡θ, for 0≤θ≤π. → 8π

Questions about these worksheets

Yes. Every arc length of parametric and polar curves 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.

Calculus is usually taken around 12th grade or a first college course, 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 arc length of parametric and polar curves 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: parametric curves and polar curves. Tick as many as you want and set how many of each, or let it spread them evenly.

Leaving out r² in the polar formula and integrating |dr/dθ| alone. A circle r = 3 has dr/dθ = 0 and still has length: the r² term is the part of the motion that goes around.