Polar calculus worksheet

A polar curve is a parametric curve in disguise: x = r cos θ and y = r sin θ, so dy/dx is their derivatives divided. Area uses thin wedges instead of rectangles: one half of r² dθ.

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Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name Calculus with Polar Curves Date Period Answer each question. Give exact values. Find $\frac{d y}{d x}$ for $r = 6 \sin θ$ at $θ = \frac{π}{3}$ Find the area inside $r = 5 + 2 \cos θ$ Find $\frac{d y}{d x}$ for $r = 3 (1 - \cos θ)$ at $θ = \frac{π}{4}$ Find the area of one petal of $r = 5 \cos 3 θ$ Find $\frac{d y}{d x}$ for $r = 1 + \cos θ$ at $θ = \frac{π}{4}$ Find the area of one petal of $r = \cos 2 θ$ Find $\frac{d y}{d x}$ for $r = 2 (1 - \cos θ)$ at $θ = \frac{π}{4}$ Find the area inside $r = 1 + \cos θ$ Find $\frac{d y}{d x}$ for $r = 2 (1 + \cos θ)$ at $θ = \frac{π}{2}$ Find the area inside $r = 2 (1 + \cos θ)$ Find $\frac{d y}{d x}$ for $r = 3 (1 + \cos θ)$ at $θ = \frac{π}{2}$ Find the area inside $r = 2 \sin θ$ Find $\frac{d y}{d x}$ for $r = 2 (1 - \cos θ)$ at $θ = \frac{2 π}{3}$ Find the area inside $r = 2 + \cos θ$ Find $\frac{d y}{d x}$ for $r = 2 (1 + \cos θ)$ at $θ = \frac{π}{4}$ Find the area inside $r = 3 (1 + \cos θ)$ Find $\frac{d y}{d x}$ for $r = 4 \sin θ$ at $θ = \frac{π}{3}$ Find the area inside $r = 5 (1 + \cos θ)$ Find $\frac{d y}{d x}$ for $r = 4 (1 - \cos θ)$ at $θ = \frac{π}{4}$ Find the area inside $r = 4 \sin θ$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Calculus with Polar Curves

Date Period

Answer each question. Give exact values.

Find $\frac{d y}{d x}$ for $r = 6 \sin θ$ at $θ = \frac{π}{3}$
Find the area inside $r = 5 + 2 \cos θ$
Find $\frac{d y}{d x}$ for $r = 3 (1 - \cos θ)$ at $θ = \frac{π}{4}$
Find the area of one petal of $r = 5 \cos 3 θ$
Find $\frac{d y}{d x}$ for $r = 1 + \cos θ$ at $θ = \frac{π}{4}$
Find the area of one petal of $r = \cos 2 θ$
Find $\frac{d y}{d x}$ for $r = 2 (1 - \cos θ)$ at $θ = \frac{π}{4}$
Find the area inside $r = 1 + \cos θ$
Find $\frac{d y}{d x}$ for $r = 2 (1 + \cos θ)$ at $θ = \frac{π}{2}$
Find the area inside $r = 2 (1 + \cos θ)$
Find $\frac{d y}{d x}$ for $r = 3 (1 + \cos θ)$ at $θ = \frac{π}{2}$
Find the area inside $r = 2 \sin θ$
Find $\frac{d y}{d x}$ for $r = 2 (1 - \cos θ)$ at $θ = \frac{2 π}{3}$
Find the area inside $r = 2 + \cos θ$
Find $\frac{d y}{d x}$ for $r = 2 (1 + \cos θ)$ at $θ = \frac{π}{4}$
Find the area inside $r = 3 (1 + \cos θ)$
Find $\frac{d y}{d x}$ for $r = 4 \sin θ$ at $θ = \frac{π}{3}$
Find the area inside $r = 5 (1 + \cos θ)$
Find $\frac{d y}{d x}$ for $r = 4 (1 - \cos θ)$ at $θ = \frac{π}{4}$
Find the area inside $r = 4 \sin θ$



Sigma Prep · sigmaprep.io/worksheets Name Calculus with Polar CurvesAnswer keyVersion 1 Date Period Answer each question. Give exact values. Find $\frac{d y}{d x}$ for $r = 6 \sin θ$ at $θ = \frac{π}{3}$ $- \sqrt{3}$ Find the area inside $r = 5 + 2 \cos θ$ $27 π$ Find $\frac{d y}{d x}$ for $r = 3 (1 - \cos θ)$ at $θ = \frac{π}{4}$ $1 + \sqrt{2}$ Find the area of one petal of $r = 5 \cos 3 θ$ $\frac{25 π}{12}$ Find $\frac{d y}{d x}$ for $r = 1 + \cos θ$ at $θ = \frac{π}{4}$ $1 - \sqrt{2}$ Find the area of one petal of $r = \cos 2 θ$ $\frac{π}{8}$ Find $\frac{d y}{d x}$ for $r = 2 (1 - \cos θ)$ at $θ = \frac{π}{4}$ $1 + \sqrt{2}$ Find the area inside $r = 1 + \cos θ$ $\frac{3 π}{2}$ Find $\frac{d y}{d x}$ for $r = 2 (1 + \cos θ)$ at $θ = \frac{π}{2}$ $1$ Find the area inside $r = 2 (1 + \cos θ)$ $6 π$ Find $\frac{d y}{d x}$ for $r = 3 (1 + \cos θ)$ at $θ = \frac{π}{2}$ $1$ Find the area inside $r = 2 \sin θ$ $π$ Find $\frac{d y}{d x}$ for $r = 2 (1 - \cos θ)$ at $θ = \frac{2 π}{3}$ $0$ Find the area inside $r = 2 + \cos θ$ $\frac{9 π}{2}$ Find $\frac{d y}{d x}$ for $r = 2 (1 + \cos θ)$ at $θ = \frac{π}{4}$ $1 - \sqrt{2}$ Find the area inside $r = 3 (1 + \cos θ)$ $\frac{27 π}{2}$ Find $\frac{d y}{d x}$ for $r = 4 \sin θ$ at $θ = \frac{π}{3}$ $- \sqrt{3}$ Find the area inside $r = 5 (1 + \cos θ)$ $\frac{75 π}{2}$ Find $\frac{d y}{d x}$ for $r = 4 (1 - \cos θ)$ at $θ = \frac{π}{4}$ $1 + \sqrt{2}$ Find the area inside $r = 4 \sin θ$ $4 π$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Calculus with Polar CurvesAnswer keyVersion 1

Date Period

Answer each question. Give exact values.

Find $\frac{d y}{d x}$ for $r = 6 \sin θ$ at $θ = \frac{π}{3}$ $- \sqrt{3}$
Find the area inside $r = 5 + 2 \cos θ$ $27 π$
Find $\frac{d y}{d x}$ for $r = 3 (1 - \cos θ)$ at $θ = \frac{π}{4}$ $1 + \sqrt{2}$
Find the area of one petal of $r = 5 \cos 3 θ$ $\frac{25 π}{12}$
Find $\frac{d y}{d x}$ for $r = 1 + \cos θ$ at $θ = \frac{π}{4}$ $1 - \sqrt{2}$
Find the area of one petal of $r = \cos 2 θ$ $\frac{π}{8}$
Find $\frac{d y}{d x}$ for $r = 2 (1 - \cos θ)$ at $θ = \frac{π}{4}$ $1 + \sqrt{2}$
Find the area inside $r = 1 + \cos θ$ $\frac{3 π}{2}$
Find $\frac{d y}{d x}$ for $r = 2 (1 + \cos θ)$ at $θ = \frac{π}{2}$ $1$
Find the area inside $r = 2 (1 + \cos θ)$ $6 π$
Find $\frac{d y}{d x}$ for $r = 3 (1 + \cos θ)$ at $θ = \frac{π}{2}$ $1$
Find the area inside $r = 2 \sin θ$ $π$
Find $\frac{d y}{d x}$ for $r = 2 (1 - \cos θ)$ at $θ = \frac{2 π}{3}$ $0$
Find the area inside $r = 2 + \cos θ$ $\frac{9 π}{2}$
Find $\frac{d y}{d x}$ for $r = 2 (1 + \cos θ)$ at $θ = \frac{π}{4}$ $1 - \sqrt{2}$
Find the area inside $r = 3 (1 + \cos θ)$ $\frac{27 π}{2}$
Find $\frac{d y}{d x}$ for $r = 4 \sin θ$ at $θ = \frac{π}{3}$ $- \sqrt{3}$
Find the area inside $r = 5 (1 + \cos θ)$ $\frac{75 π}{2}$
Find $\frac{d y}{d x}$ for $r = 4 (1 - \cos θ)$ at $θ = \frac{π}{4}$ $1 + \sqrt{2}$
Find the area inside $r = 4 \sin θ$ $4 π$

How to do these

Find the area inside $r = 2 (1 + \cos θ)$

Area is one half of the integral of r² from 0 to 2π.
r² = 4(1 + 2cos θ + cos²θ), whose integral over a full turn is 4(2π + 0 + π) = 12π.
Half of that.

The answer is $6 π$ .

Where students go wrong

Integrating over the wrong range of θ. A rose petal is traced over a small interval between two zeros of r; integrating over the whole turn counts every petal, and sometimes counts them twice.

Also called polar area, area of a polar curve, polar derivative or dy/dx polar.

What you can put on this worksheet

dy/dx for a polar curve: Find $\frac{d y}{d x}$ for $r = 6 \cos θ$ at $θ = \frac{π}{3}$ → $\frac{\sqrt{3}}{3}$
Area inside a polar curve: Find the area inside $r = 6 \sin θ$ → $9 π$

Questions about these worksheets

Yes. Every polar calculus 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 polar calculus 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: dy/dx for a polar curve and area inside a polar curve. Tick as many as you want and set how many of each, or let it spread them evenly.

Integrating over the wrong range of θ. A rose petal is traced over a small interval between two zeros of r; integrating over the whole turn counts every petal, and sometimes counts them twice.

Polar calculus worksheet

Select your difficulty

Calculus with Polar Curves

Calculus with Polar CurvesAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these polar calculus worksheets free?+

Do the polar calculus worksheets come with an answer key?+

Are these polar calculus worksheets printable?+

What grade are these polar calculus worksheets for?+

Can I get a different version of the same polar calculus worksheet?+

Is this a free alternative to Kuta Software for polar calculus?+

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What do students get wrong with polar calculus?+