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 r from the origin and an angle θ from the positive x-axis, written as (r,θ).

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=2 to polar form, simplified.

Example 2

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

Example 3

easy
Convert (r,θ)=(2,π3) to Cartesian.

Example 4

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

Example 5

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

Example 6

medium
Find the rectangular coordinates of the polar point (−2,π4).

Example 7

easy
Convert the polar point (r,θ)=(4,2π3) to rectangular coordinates.

Example 8

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

Example 9

easy
Convert the polar point (r,θ)=(2,5π6) to rectangular coordinates.

Example 10

medium
Give two other polar representations of (2,π6).

Example 11

easy
Convert (r,θ)=(5,3π2) to Cartesian.

Example 12

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

Example 13

medium
Find the distance between the polar points (3,π6) and (5,π2).

Example 14

challenge
The polar curve r=2+2cos⁡θ and the circle r=3 intersect. Find all intersection points in polar form with 0≤θ<2π.

Example 15

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

Example 16

easy
Convert the polar point (r,θ)=(3,π6) to rectangular coordinates.

Example 17

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

Example 18

medium
Convert the polar equation r=−3csc⁡θ to a rectangular equation.

Example 19

medium
Convert the rectangular point (−2,−23) to polar coordinates with r>0 and 0≤θ<2π.

Example 20

easy
Convert (r,θ)=(2,π2) to Cartesian.
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