On [0, 2π), find where r = 0 and the maximum of |r| for; r = 2sqrt(3) + 4sin theta

r = 0 at theta = 4π/3, 5π/3; [10pt] max |r| = 4 + 2sqrt(3)

On [0, 2π), find where r = 0 and the maximum of |r| for; r = 2 + 4cos theta

r = 0 at theta = 2π/3, 4π/3; [10pt] max |r| = 6

On [0, 2π), find where r = 0 and the maximum of |r| for; r = sqrt(2) - 2sin theta

r = 0 at theta = π/4, 3π/4; [10pt] max |r| = 2 + sqrt(2)

On [0, 2π), find where r = 0 and the maximum of |r| for; r = 5 - 10sin theta

r = 0 at theta = π/6, 5π/6; [10pt] max |r| = 15

On [0, 2π), find where r = 0 and the maximum of |r| for; r = sqrt(3) - 2cos theta

r = 0 at theta = π/6, 11π/6; [10pt] max |r| = 2 + sqrt(3)

On [0, 2π), find where r = 0 and the maximum of |r| for; r = 4 - 8cos theta

r = 0 at theta = π/3, 5π/3; [10pt] max |r| = 12

On [0, 2π), find where r = 0 and the maximum of |r| for; r = 2sqrt(2) - 4cos theta

r = 0 at theta = π/4, 7π/4; [10pt] max |r| = 4 + 2sqrt(2)

On [0, 2π), find where r = 0 and the maximum of |r| for; r = 3 + 6sin theta

r = 0 at theta = 7π/6, 11π/6; [10pt] max |r| = 9

On [0, 2π), find where r = 0 and the maximum of |r| for; r = 3sqrt(2) + 6cos theta

r = 0 at theta = 3π/4, 5π/4; [10pt] max |r| = 6 + 3sqrt(2)

On [0, 2π), find where r = 0 and the maximum of |r| for; r = 3sqrt(3) - 6sin theta

r = 0 at theta = π/3, 2π/3; [10pt] max |r| = 6 + 3sqrt(3)

On [0, 2π), find where r = 0 and the maximum of |r| for; r = 5sin 2theta

r = 0 at theta = 0, π/2, π, 3π/2; [10pt] max |r| = 5

Which symmetries does the graph of r = 5sin 2theta have: the polar axis, the line theta = π/2, the pole?

the polar axis, the line theta = π/2, the pole

For r = 4sin 4theta on [0, π), find where r = 0 and where |r| is largest.

r = 0 at theta = 0, π/4, π/2, 3π/4; |r| is largest, 4, at theta = π/8, 3π/8, 5π/8, 7π/8

Worksheets · Precalculus

Polar graphs worksheet

Q: Are these polar graphs worksheets free?

Yes. Every polar graphs 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.

Q: Do the polar graphs worksheets come with an answer key?

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.

Q: Are these polar graphs worksheets printable?

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.

Q: What grade are these polar graphs worksheets for?

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.

Q: Can I get a different version of the same polar graphs worksheet?

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.

Q: Is this a free alternative to Kuta Software for polar graphs?

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 graphs 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.

Q: Can I choose how hard the polar graphs questions are?

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.

Q: What kinds of question are on a polar graphs worksheet?

You pick from 6: name the graph from the equation, graph it on a polar grid, where r is zero, and the largest r, name the curve from its graph, symmetry and zeros and the largest |r|, and where. Tick as many as you want and set how many of each, or let it spread them evenly.

Q: What do students get wrong with polar graphs?

Counting n petals for every rose. r = cos 2θ has four petals, not two; only odd n gives n petals.

The shape is in the form of the equation. r = a cos θ is a circle; r = a cos nθ is a rose with n petals when n is odd and 2n when it is even; r = a + b cos θ is a limaçon, a cardioid when a = b and looped when a < b; r² = a² cos 2θ is a lemniscate. Sine versions are the same shapes turned. The other half of each sheet asks where r is zero on [0, 2π) and how large |r| gets.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Polar Graphs Date Period Answer each question. In naming a rose give its number of petals, and for a limaçon say whether it has an inner loop. Name the polar graph from its equation Name the graph: $r = 4 + 3 \cos θ$ Name the graph: $r = 3 \cos (2 θ)$ Name the graph: $r = 5 + 5 \sin θ$ Name the graph: $r = 5 \sin (3 θ)$ Name the graph: $r = θ$ Name the graph: $r = \cos θ$ Name the graph: $r^{2} = \sin (2 θ)$ Name the graph: $r = 1 + \sin θ$ Name the graph: $r = \sin (4 θ)$ Name the graph: $r = 3 + \cos θ$ Find the zeros and the maximum $\|r\|$ On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for $r = 2 \sqrt{3} + 4 \sin θ$ On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for $r = 2 + 4 \cos θ$ On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for $r = \sqrt{2} - 2 \sin θ$ On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for $r = 5 - 10 \sin θ$ On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for $r = \sqrt{3} - 2 \cos θ$ On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for $r = 4 - 8 \cos θ$ On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for $r = 2 \sqrt{2} - 4 \cos θ$ On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for $r = 3 + 6 \sin θ$ On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for $r = 3 \sqrt{2} + 6 \cos θ$ On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for $r = 3 \sqrt{3} - 6 \sin θ$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Polar Graphs

Date Period

Answer each question. In naming a rose give its number of petals, and for a limaçon say whether it has an inner loop.

Name the polar graph from its equation
Name the graph: $r = 4 + 3 \cos θ$
Name the graph: $r = 3 \cos (2 θ)$
Name the graph: $r = 5 + 5 \sin θ$
Name the graph: $r = 5 \sin (3 θ)$
Name the graph: $r = θ$
Name the graph: $r = \cos θ$
Name the graph: $r^{2} = \sin (2 θ)$
Name the graph: $r = 1 + \sin θ$
Name the graph: $r = \sin (4 θ)$
Name the graph: $r = 3 + \cos θ$

Find the zeros and the maximum $|r|$
On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = 2 \sqrt{3} + 4 \sin θ$
On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = 2 + 4 \cos θ$
On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = \sqrt{2} - 2 \sin θ$
On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = 5 - 10 \sin θ$
On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = \sqrt{3} - 2 \cos θ$
On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = 4 - 8 \cos θ$
On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = 2 \sqrt{2} - 4 \cos θ$
On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = 3 + 6 \sin θ$
On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = 3 \sqrt{2} + 6 \cos θ$
On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = 3 \sqrt{3} - 6 \sin θ$



Sigma Prep · sigmaprep.io/worksheets Name Polar GraphsAnswer keyVersion 1 Date Period Answer each question. In naming a rose give its number of petals, and for a limaçon say whether it has an inner loop. Name the polar graph from its equation Name the graph: $r = 4 + 3 \cos θ$ Limaçon, no inner loop Name the graph: $r = 3 \cos (2 θ)$ Rose, $4$ petals Name the graph: $r = 5 + 5 \sin θ$ Cardioid Name the graph: $r = 5 \sin (3 θ)$ Rose, $3$ petals Name the graph: $r = θ$ Spiral Name the graph: $r = \cos θ$ Circle Name the graph: $r^{2} = \sin (2 θ)$ Lemniscate Name the graph: $r = 1 + \sin θ$ Cardioid Name the graph: $r = \sin (4 θ)$ Rose, $8$ petals Name the graph: $r = 3 + \cos θ$ Limaçon, no inner loop Find the zeros and the maximum $\|r\|$ On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for $r = 2 \sqrt{3} + 4 \sin θ$ $\begin{array}{ll} r = 0 at θ = & \frac{4 π}{3}, \frac{5 π}{3} \\ max ∣ r ∣ = 4 + 2 \sqrt{3} \end{array}$ On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for $r = 2 + 4 \cos θ$ $\begin{array}{ll} r = 0 at θ = & \frac{2 π}{3}, \frac{4 π}{3} \\ max ∣ r ∣ = 6 \end{array}$ On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for $r = \sqrt{2} - 2 \sin θ$ $\begin{array}{ll} r = 0 at θ = & \frac{π}{4}, \frac{3 π}{4} \\ max ∣ r ∣ = 2 + \sqrt{2} \end{array}$ On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for $r = 5 - 10 \sin θ$ $\begin{array}{ll} r = 0 at θ = & \frac{π}{6}, \frac{5 π}{6} \\ max ∣ r ∣ = 15 \end{array}$ On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for $r = \sqrt{3} - 2 \cos θ$ $\begin{array}{ll} r = 0 at θ = & \frac{π}{6}, \frac{11 π}{6} \\ max ∣ r ∣ = 2 + \sqrt{3} \end{array}$ On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for $r = 4 - 8 \cos θ$ $\begin{array}{ll} r = 0 at θ = & \frac{π}{3}, \frac{5 π}{3} \\ max ∣ r ∣ = 12 \end{array}$ On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for $r = 2 \sqrt{2} - 4 \cos θ$ $\begin{array}{ll} r = 0 at θ = & \frac{π}{4}, \frac{7 π}{4} \\ max ∣ r ∣ = 4 + 2 \sqrt{2} \end{array}$ On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for $r = 3 + 6 \sin θ$ $\begin{array}{ll} r = 0 at θ = & \frac{7 π}{6}, \frac{11 π}{6} \\ max ∣ r ∣ = 9 \end{array}$ On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for $r = 3 \sqrt{2} + 6 \cos θ$ $\begin{array}{ll} r = 0 at θ = & \frac{3 π}{4}, \frac{5 π}{4} \\ max ∣ r ∣ = 6 + 3 \sqrt{2} \end{array}$ On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for $r = 3 \sqrt{3} - 6 \sin θ$ $\begin{array}{ll} r = 0 at θ = & \frac{π}{3}, \frac{2 π}{3} \\ max ∣ r ∣ = 6 + 3 \sqrt{3} \end{array}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Polar GraphsAnswer keyVersion 1

Date Period

Answer each question. In naming a rose give its number of petals, and for a limaçon say whether it has an inner loop.

Name the polar graph from its equation
Name the graph: $r = 4 + 3 \cos θ$ Limaçon, no inner loop
Name the graph: $r = 3 \cos (2 θ)$ Rose, $4$ petals
Name the graph: $r = 5 + 5 \sin θ$ Cardioid
Name the graph: $r = 5 \sin (3 θ)$ Rose, $3$ petals
Name the graph: $r = θ$ Spiral
Name the graph: $r = \cos θ$ Circle
Name the graph: $r^{2} = \sin (2 θ)$ Lemniscate
Name the graph: $r = 1 + \sin θ$ Cardioid
Name the graph: $r = \sin (4 θ)$ Rose, $8$ petals
Name the graph: $r = 3 + \cos θ$ Limaçon, no inner loop

Find the zeros and the maximum $|r|$
On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = 2 \sqrt{3} + 4 \sin θ$ $\begin{array}{ll} r = 0 at θ = & \frac{4 π}{3}, \frac{5 π}{3} \\ max ∣ r ∣ = 4 + 2 \sqrt{3} \end{array}$
On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = 2 + 4 \cos θ$ $\begin{array}{ll} r = 0 at θ = & \frac{2 π}{3}, \frac{4 π}{3} \\ max ∣ r ∣ = 6 \end{array}$
On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = \sqrt{2} - 2 \sin θ$ $\begin{array}{ll} r = 0 at θ = & \frac{π}{4}, \frac{3 π}{4} \\ max ∣ r ∣ = 2 + \sqrt{2} \end{array}$
On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = 5 - 10 \sin θ$ $\begin{array}{ll} r = 0 at θ = & \frac{π}{6}, \frac{5 π}{6} \\ max ∣ r ∣ = 15 \end{array}$
On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = \sqrt{3} - 2 \cos θ$ $\begin{array}{ll} r = 0 at θ = & \frac{π}{6}, \frac{11 π}{6} \\ max ∣ r ∣ = 2 + \sqrt{3} \end{array}$
On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = 4 - 8 \cos θ$ $\begin{array}{ll} r = 0 at θ = & \frac{π}{3}, \frac{5 π}{3} \\ max ∣ r ∣ = 12 \end{array}$
On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = 2 \sqrt{2} - 4 \cos θ$ $\begin{array}{ll} r = 0 at θ = & \frac{π}{4}, \frac{7 π}{4} \\ max ∣ r ∣ = 4 + 2 \sqrt{2} \end{array}$
On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = 3 + 6 \sin θ$ $\begin{array}{ll} r = 0 at θ = & \frac{7 π}{6}, \frac{11 π}{6} \\ max ∣ r ∣ = 9 \end{array}$
On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = 3 \sqrt{2} + 6 \cos θ$ $\begin{array}{ll} r = 0 at θ = & \frac{3 π}{4}, \frac{5 π}{4} \\ max ∣ r ∣ = 6 + 3 \sqrt{2} \end{array}$
On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = 3 \sqrt{3} - 6 \sin θ$ $\begin{array}{ll} r = 0 at θ = & \frac{π}{3}, \frac{2 π}{3} \\ max ∣ r ∣ = 6 + 3 \sqrt{3} \end{array}$

How to do these

Name the graph: $r = 2 + 5 \sin θ$

a + b sin θ is a limaçon.
a = 2 is less than b = 5, so it has an inner loop.

The answer is Limaçon with an inner loop.

Where students go wrong

Counting n petals for every rose. r = cos 2θ has four petals, not two; only odd n gives n petals.

Also called polar equations, rose curves, limacon, cardioid, lemniscate or graphing polar equations.

What you can put on this worksheet

Name the graph from the equation: Name the graph: $r = 5 + 5 \sin θ$ → Cardioid
Graph it on a polar grid: Graph $r = 3 + 3 \sin θ$ and name the curve. — a blank polar grid → Cardioid
Where r is zero, and the largest r: On $[0, 2 π)$ , find where $r = 0$ and the maximum of $∣ r ∣$ for
$r = 5 \sin 2 θ$ → $\begin{array}{ll} r = 0 at θ = & 0, \frac{π}{2}, π, \frac{3 π}{2} \\ max ∣ r ∣ = 5 \end{array}$
Name the curve from its graph: Name the polar curve shown. — the graph of cardioid on a polar grid → Cardioid
Symmetry: Which symmetries does the graph of $r = 5 \sin 2 θ$ have: the polar axis, the line $θ = \frac{π}{2}$ , the pole? → the polar axis, the line $θ = \frac{π}{2}$ , the pole
Zeros and the largest |r|, and where: For $r = 4 \sin 4 θ$ on $[0, π)$ , find where $r = 0$ and where $∣ r ∣$ is largest. → $r = 0$ at $θ = 0, \frac{π}{4}, \frac{π}{2}, \frac{3 π}{4}$
$∣ r ∣$ is largest, $4$ , at $θ = \frac{π}{8}, \frac{3 π}{8}, \frac{5 π}{8}, \frac{7 π}{8}$

Questions about these worksheets