Practice Polar Coordinates in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A coordinate system where each point in the plane is described by a distance rr from the origin and an angle θ\theta from the positive xx-axis, written as (r,θ)(r, \theta).

Instead of 'go right 3, up 4' (Cartesian), polar says 'go 5 units in the direction of 53°.' It's how a radar works—distance and direction from a central point. Some shapes that look complicated in Cartesian coordinates become beautifully simple in polar.

Showing a random 20 of 50 problems.

Example 1

hard
Convert the rectangular equation xy=2xy = 2 to polar form, simplified.

Example 2

medium
Convert the line y=xy = x to a polar equation.

Example 3

easy
Convert (r,θ)=(2,π3)(r,\theta)=(2,\frac{\pi}{3}) to Cartesian.

Example 4

hard
Convert the polar equation r=2sin⁡θ+2cos⁡θr = 2\sin\theta + 2\cos\theta to a rectangular equation and identify the curve.

Example 5

challenge
Convert r=2sin⁡θ+2cos⁡θr = 2\sin\theta + 2\cos\theta to Cartesian and identify the curve.

Example 6

medium
Find the rectangular coordinates of the polar point (−2,π4)(-2, \frac{\pi}{4}).

Example 7

easy
Convert the polar point (r,θ)=(4,2π3)(r,\theta) = (4, \frac{2\pi}{3}) to rectangular coordinates.

Example 8

easy
Convert the polar point (r,θ)=(4,0)(r,\theta)=(4,0) to Cartesian.

Example 9

easy
Convert the polar point (r,θ)=(2,5π6)(r,\theta) = (2, \frac{5\pi}{6}) to rectangular coordinates.

Example 10

medium
Give two other polar representations of (2,π6)(2,\frac{\pi}{6}).

Example 11

easy
Convert (r,θ)=(5,3π2)(r,\theta)=(5,\frac{3\pi}{2}) to Cartesian.

Example 12

medium
Convert the rectangular equation y=xy = x to polar form.

Example 13

medium
Find the distance between the polar points (3,π6)(3, \frac{\pi}{6}) and (5,π2)(5, \frac{\pi}{2}).

Example 14

challenge
The polar curve r=2+2cos⁡θr = 2 + 2\cos\theta and the circle r=3r = 3 intersect. Find all intersection points in polar form with 0≤θ<2π0 \le \theta < 2\pi.

Example 15

medium
Find the angle θ\theta (in [0,2π)[0,2\pi)) for the Cartesian point (−2,−2)(-2,-2).

Example 16

easy
Convert the polar point (r,θ)=(3,π6)(r,\theta) = (3, \frac{\pi}{6}) to rectangular coordinates.

Example 17

medium
Convert the rectangular equation x2+y2=4yx^2 + y^2 = 4y to polar form.

Example 18

medium
Convert the polar equation r=−3csc⁡θr = -3\csc\theta to a rectangular equation.

Example 19

medium
Convert the rectangular point (−2,−23)(-2, -2\sqrt{3}) to polar coordinates with r>0r > 0 and 0≤θ<2π0 \le \theta < 2\pi.

Example 20

easy
Convert (r,θ)=(2,π2)(r,\theta)=(2,\frac{\pi}{2}) to Cartesian.
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