Practice Polar Graphs in Math

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

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Quick Recap

Graphs of equations in the form r=f(θ)r = f(\theta), producing curves such as rose curves, cardioids, limaçons, and circles in the polar plane.

As the angle θ\theta sweeps around, the distance rr changes according to the equation, tracing out a curve. Think of it like a radar sweep where the blip's distance from the center varies with direction. This creates curves with stunning symmetry that would require complex implicit equations in Cartesian coordinates.

Showing a random 20 of 50 problems.

Example 1

medium
Convert the polar curve r=6sin⁡θr = 6\sin\theta to rectangular form.

Example 2

medium
Identify the curve r2=9cos⁡(2θ)r^2 = 9\cos(2\theta).

Example 3

hard
Find all polar-coordinate intersection points of r=2cos⁡θr = 2\cos\theta and r=2sin⁡θr = 2\sin\theta for 0≤θ<2π0 \le \theta < 2\pi.

Example 4

hard
Find the slope of the tangent to r=4cos⁡θr = 4\cos\theta at θ=π/4\theta = \pi/4.

Example 5

easy
What curve is r=4cos⁡θr = 4\cos\theta?

Example 6

easy
How many petals does r=sin⁡(3θ)r = \sin(3\theta) have?

Example 7

medium
Find the slope of the tangent to r=1+cos⁡θr = 1 + \cos\theta at θ=π/2\theta = \pi/2.

Example 8

medium
Classify r=4+4sin⁡θr = 4 + 4\sin\theta.

Example 9

easy
Describe the graph of r=3r = 3 in polar coordinates.

Example 10

easy
How many petals does r=cos⁡(2θ)r = \cos(2\theta) have?

Example 11

challenge
For the rose r=2sin⁡(2θ)r = 2\sin(2\theta), find the area of one petal using A=12∫r2 dθA=\frac12\int r^2\,d\theta.

Example 12

easy
What curve is represented by r=−6sin⁡θr = -6\sin\theta?

Example 13

easy
What type of curve is r=3+3cos⁡θr = 3 + 3\cos\theta?

Example 14

easy
Identify r=3−5sin⁡θr = 3 - 5\sin\theta as a cardioid, dimpled limaçon, convex limaçon, or limaçon with inner loop.

Example 15

medium
Identify whether the graph of r=4sin⁡(3θ)r = 4\sin(3\theta) has any symmetry about the line θ=π/2\theta = \pi/2.

Example 16

medium
How many petals does r=3sin⁡(4θ)r = 3\sin(4\theta) have, and how long is each?

Example 17

medium
How many petals does r=2cos⁡(5θ)r = 2\cos(5\theta) have?

Example 18

hard
Find the arc length of the cardioid r=1−cos⁡θr = 1 - \cos\theta.

Example 19

easy
What curve is θ=π4\theta = \frac{\pi}{4}?

Example 20

challenge
Find the area inside the cardioid r=1+cos⁡θr = 1 + \cos\theta but outside the circle r=1r = 1.
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