Plot A(3, 0) and B(4, 5π/3)[-1.15em]0pt2.75em on a polar grid. — a blank polar grid

A: 3 out along 0; B: 4 out along 5π/3

Plot A(2, 3π/2)[-1.15em]0pt2.75em and B(1, 2π/3)[-1.15em]0pt2.75em on a polar grid. — a blank polar grid

A: 2 out along 3π/2; B: 1 out along 2π/3

Plot A(3, π/3)[-1.15em]0pt2.75em and B(1, π/3)[-1.15em]0pt2.75em on a polar grid. — a blank polar grid

A: 3 out along π/3; B: 1 out along π/3

Plot A(1, 7π/6)[-1.15em]0pt2.75em and B(1, π/6)[-1.15em]0pt2.75em on a polar grid. — a blank polar grid

A: 1 out along 7π/6; B: 1 out along π/6

Plot A(2, π/2)[-1.15em]0pt2.75em and B(3, 7π/6)[-1.15em]0pt2.75em on a polar grid. — a blank polar grid

A: 2 out along π/2; B: 3 out along 7π/6

Plot A(3, π) and B(1, 2π/3)[-1.15em]0pt2.75em on a polar grid. — a blank polar grid

A: 3 out along π; B: 1 out along 2π/3

Plot A(3, π) and B(3, 0) on a polar grid. — a blank polar grid

A: 3 out along π; B: 3 out along 0

Write all polar coordinate pairs for (6, 270°).

(6, 270° + 360°n) and (-6, 90° + 360°n), n an integer

Plot A(2, 30°) and B(3, -300°) on a polar grid. — a blank polar grid, labeled 1, 2, 3, 4

A: 2 out along 30°; B: 3 out along 60°

Worksheets · Precalculus

Polar coordinates worksheet

Q: Are these polar coordinates worksheets free?

Yes. Every polar coordinates 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 coordinates 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 coordinates 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 coordinates 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 coordinates 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 coordinates?

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 coordinates 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 coordinates questions are?

Yes. There are two 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 coordinates worksheet?

You pick from 7: polar to rectangular, rectangular to polar, another pair for the same point, plot points on a polar grid, all polar pairs for a point, distance between polar points and plot in degrees, negative angles and r. 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 coordinates?

Plotting a negative r on the angle's own ray. A negative r points the opposite way, so (-3, 4π/3) is 3 out along π/3, half a turn round from 4π/3.

A polar point says how far out and which way round, instead of how far across and how far up. Converting is just the two right-triangle relationships, so every angle here is one you already know exactly.

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 Coordinates Date Period Convert or plot each point, or answer as asked. Leave answers exact unless told to round. Convert polar to rectangular Convert to rectangular: $(2, 225^{\circ})$ Convert to rectangular: $(10, 120^{\circ})$ Convert to rectangular: $(18, 330^{\circ})$ Convert to rectangular: $(60, 30^{\circ})$ Convert to rectangular: $(34, 60^{\circ})$ Convert to rectangular: $(24, 240^{\circ})$ Convert to rectangular: $(54, 300^{\circ})$ Convert rectangular to polar Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(2 \sqrt{2}, - 2 \sqrt{2})$ Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(- 4 \sqrt{2}, 4 \sqrt{2})$ Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(3 \sqrt{2}, 3 \sqrt{2})$ Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(\sqrt{2}, - \sqrt{2})$ Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(- 5 \sqrt{2}, - 5 \sqrt{2})$ Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(4 \sqrt{2}, 4 \sqrt{2})$ Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(- 2 \sqrt{2}, 2 \sqrt{2})$ Plot points on a polar grid Plot $A (3, 0)$ and $B ⁣ (4, \frac{5 π}{3})$ on a polar grid. Plot $A ⁣ (2, \frac{3 π}{2})$ and $B ⁣ (1, \frac{2 π}{3})$ on a polar grid. Plot $A ⁣ (3, \frac{π}{3})$ and $B ⁣ (1, \frac{π}{3})$ on a polar grid. Plot $A ⁣ (1, \frac{7 π}{6})$ and $B ⁣ (1, \frac{π}{6})$ on a polar grid. Plot $A ⁣ (2, \frac{π}{2})$ and $B ⁣ (3, \frac{7 π}{6})$ on a polar grid. Plot $A (3, π)$ and $B ⁣ (1, \frac{2 π}{3})$ on a polar grid. Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Polar Coordinates

Date Period

Convert or plot each point, or answer as asked. Leave answers exact unless told to round.

Convert polar to rectangular
Convert to rectangular: $(2, 225^{\circ})$
Convert to rectangular: $(10, 120^{\circ})$
Convert to rectangular: $(18, 330^{\circ})$
Convert to rectangular: $(60, 30^{\circ})$
Convert to rectangular: $(34, 60^{\circ})$
Convert to rectangular: $(24, 240^{\circ})$
Convert to rectangular: $(54, 300^{\circ})$

Convert rectangular to polar
Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(2 \sqrt{2}, - 2 \sqrt{2})$
Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(- 4 \sqrt{2}, 4 \sqrt{2})$
Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(3 \sqrt{2}, 3 \sqrt{2})$
Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(\sqrt{2}, - \sqrt{2})$
Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(- 5 \sqrt{2}, - 5 \sqrt{2})$
Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(4 \sqrt{2}, 4 \sqrt{2})$
Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(- 2 \sqrt{2}, 2 \sqrt{2})$

Plot points on a polar grid
Plot $A (3, 0)$ and $B ⁣ (4, \frac{5 π}{3})$ on a polar grid.
Plot $A ⁣ (2, \frac{3 π}{2})$ and $B ⁣ (1, \frac{2 π}{3})$ on a polar grid.
Plot $A ⁣ (3, \frac{π}{3})$ and $B ⁣ (1, \frac{π}{3})$ on a polar grid.
Plot $A ⁣ (1, \frac{7 π}{6})$ and $B ⁣ (1, \frac{π}{6})$ on a polar grid.
Plot $A ⁣ (2, \frac{π}{2})$ and $B ⁣ (3, \frac{7 π}{6})$ on a polar grid.
Plot $A (3, π)$ and $B ⁣ (1, \frac{2 π}{3})$ on a polar grid.



Sigma Prep · sigmaprep.io/worksheets Name Polar CoordinatesAnswer keyVersion 1 Date Period Convert or plot each point, or answer as asked. Leave answers exact unless told to round. Convert polar to rectangular Convert to rectangular: $(2, 225^{\circ})$ $(- \sqrt{2}, - \sqrt{2})$ Convert to rectangular: $(10, 120^{\circ})$ $(- 5, 5 \sqrt{3})$ Convert to rectangular: $(18, 330^{\circ})$ $(9 \sqrt{3}, - 9)$ Convert to rectangular: $(60, 30^{\circ})$ $(30 \sqrt{3}, 30)$ Convert to rectangular: $(34, 60^{\circ})$ $(17, 17 \sqrt{3})$ Convert to rectangular: $(24, 240^{\circ})$ $(- 12, - 12 \sqrt{3})$ Convert to rectangular: $(54, 300^{\circ})$ $(27, - 27 \sqrt{3})$ Convert rectangular to polar Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(2 \sqrt{2}, - 2 \sqrt{2})$ $(4, 315^{\circ})$ Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(- 4 \sqrt{2}, 4 \sqrt{2})$ $(8, 135^{\circ})$ Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(3 \sqrt{2}, 3 \sqrt{2})$ $(6, 45^{\circ})$ Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(\sqrt{2}, - \sqrt{2})$ $(2, 315^{\circ})$ Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(- 5 \sqrt{2}, - 5 \sqrt{2})$ $(10, 225^{\circ})$ Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(4 \sqrt{2}, 4 \sqrt{2})$ $(8, 45^{\circ})$ Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(- 2 \sqrt{2}, 2 \sqrt{2})$ $(4, 135^{\circ})$ Plot points on a polar grid Plot $A (3, 0)$ and $B ⁣ (4, \frac{5 π}{3})$ on a polar grid. $A$ : $3$ out along $0$ ; $B$ : $4$ out along $\frac{5 π}{3}$ Plot $A ⁣ (2, \frac{3 π}{2})$ and $B ⁣ (1, \frac{2 π}{3})$ on a polar grid. $A$ : $2$ out along $\frac{3 π}{2}$ ; $B$ : $1$ out along $\frac{2 π}{3}$ Plot $A ⁣ (3, \frac{π}{3})$ and $B ⁣ (1, \frac{π}{3})$ on a polar grid. $A$ : $3$ out along $\frac{π}{3}$ ; $B$ : $1$ out along $\frac{π}{3}$ Plot $A ⁣ (1, \frac{7 π}{6})$ and $B ⁣ (1, \frac{π}{6})$ on a polar grid. $A$ : $1$ out along $\frac{7 π}{6}$ ; $B$ : $1$ out along $\frac{π}{6}$ Plot $A ⁣ (2, \frac{π}{2})$ and $B ⁣ (3, \frac{7 π}{6})$ on a polar grid. $A$ : $2$ out along $\frac{π}{2}$ ; $B$ : $3$ out along $\frac{7 π}{6}$ Plot $A (3, π)$ and $B ⁣ (1, \frac{2 π}{3})$ on a polar grid. $A$ : $3$ out along $π$ ; $B$ : $1$ out along $\frac{2 π}{3}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Polar CoordinatesAnswer keyVersion 1

Date Period

Convert or plot each point, or answer as asked. Leave answers exact unless told to round.

Convert polar to rectangular
Convert to rectangular: $(2, 225^{\circ})$ $(- \sqrt{2}, - \sqrt{2})$
Convert to rectangular: $(10, 120^{\circ})$ $(- 5, 5 \sqrt{3})$
Convert to rectangular: $(18, 330^{\circ})$ $(9 \sqrt{3}, - 9)$
Convert to rectangular: $(60, 30^{\circ})$ $(30 \sqrt{3}, 30)$
Convert to rectangular: $(34, 60^{\circ})$ $(17, 17 \sqrt{3})$
Convert to rectangular: $(24, 240^{\circ})$ $(- 12, - 12 \sqrt{3})$
Convert to rectangular: $(54, 300^{\circ})$ $(27, - 27 \sqrt{3})$

Convert rectangular to polar
Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(2 \sqrt{2}, - 2 \sqrt{2})$ $(4, 315^{\circ})$
Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(- 4 \sqrt{2}, 4 \sqrt{2})$ $(8, 135^{\circ})$
Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(3 \sqrt{2}, 3 \sqrt{2})$ $(6, 45^{\circ})$
Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(\sqrt{2}, - \sqrt{2})$ $(2, 315^{\circ})$
Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(- 5 \sqrt{2}, - 5 \sqrt{2})$ $(10, 225^{\circ})$
Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(4 \sqrt{2}, 4 \sqrt{2})$ $(8, 45^{\circ})$
Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(- 2 \sqrt{2}, 2 \sqrt{2})$ $(4, 135^{\circ})$

Plot points on a polar grid
Plot $A (3, 0)$ and $B ⁣ (4, \frac{5 π}{3})$ on a polar grid. $A$ : $3$ out along $0$ ; $B$ : $4$ out along $\frac{5 π}{3}$
Plot $A ⁣ (2, \frac{3 π}{2})$ and $B ⁣ (1, \frac{2 π}{3})$ on a polar grid. $A$ : $2$ out along $\frac{3 π}{2}$ ; $B$ : $1$ out along $\frac{2 π}{3}$
Plot $A ⁣ (3, \frac{π}{3})$ and $B ⁣ (1, \frac{π}{3})$ on a polar grid. $A$ : $3$ out along $\frac{π}{3}$ ; $B$ : $1$ out along $\frac{π}{3}$
Plot $A ⁣ (1, \frac{7 π}{6})$ and $B ⁣ (1, \frac{π}{6})$ on a polar grid. $A$ : $1$ out along $\frac{7 π}{6}$ ; $B$ : $1$ out along $\frac{π}{6}$
Plot $A ⁣ (2, \frac{π}{2})$ and $B ⁣ (3, \frac{7 π}{6})$ on a polar grid. $A$ : $2$ out along $\frac{π}{2}$ ; $B$ : $3$ out along $\frac{7 π}{6}$
Plot $A (3, π)$ and $B ⁣ (1, \frac{2 π}{3})$ on a polar grid. $A$ : $3$ out along $π$ ; $B$ : $1$ out along $\frac{2 π}{3}$

How to do these

Convert to rectangular:
$(6, \frac{5 π}{6})$

x is 6 cos of 5 pi over 6, and cosine there is negative root 3 over 2.
That gives x equals negative 3 root 3.
y is 6 sin of 5 pi over 6, which is 6 times one half.

The answer is $(- 3 \sqrt{3}, 3)$ .

Where students go wrong

Plotting a negative r on the angle's own ray. A negative r points the opposite way, so (-3, 4π/3) is 3 out along π/3, half a turn round from 4π/3.

Also called polar and rectangular coordinates, converting coordinates or polar form of a point.

What you can put on this worksheet

Polar to rectangular: Convert to rectangular: $(34, 210^{\circ})$ → $(- 17 \sqrt{3}, - 17)$
Rectangular to polar: Convert to polar with $r > 0$ and $0^{\circ} \leq θ < 360^{\circ}$ : $(- 3 \sqrt{2}, - 3 \sqrt{2})$ → $(6, 225^{\circ})$
Another pair for the same point: Name the same point with $r > 0$ and $- 360^{\circ} \leq θ < 0^{\circ}$ : $(6, 210^{\circ})$ → $(6, - 150^{\circ})$
Plot points on a polar grid: Plot $A (3, π)$ and $B (3, 0)$ on a polar grid. — a blank polar grid → $A$ : $3$ out along $π$ ; $B$ : $3$ out along $0$
All polar pairs for a point: Write all polar coordinate pairs for $(6, 270^{\circ})$ . → $(6, 270^{\circ} + 360^{\circ} n)$ and $(- 6, 90^{\circ} + 360^{\circ} n)$ , $n an integer$
Distance between polar points: Find the distance between $(5, \frac{π}{3})$ and $(6, \frac{11 π}{6})$ , to the nearest hundredth. → $7.81$
Plot in degrees, negative angles and r: Plot $A (2, 30^{\circ})$ and $B (3, - 300^{\circ})$ on a polar grid. — a blank polar grid, labeled 1, 2, 3, 4 → A: $2$ out along $30^{\circ}$ ; B: $3$ out along $60^{\circ}$

Questions about these worksheets