Polar Graphs Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Polar Graphs.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Graphs of r=f(θ)r=f(\theta) — roses, cardioids, limaçons — drawn as the radius changes with direction.

Common stuck point: The procedure for polar graphs is the easy part; the trap is miscounting rose petals. Asking "Is the equation rr expressed in terms of θ\theta, traced by sweeping the angle around?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the equation rr expressed in terms of θ\theta, traced by sweeping the angle around?

Worked Examples

Example 1

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

Answer

A circle of radius 3 centered at the origin\text{A circle of radius } 3 \text{ centered at the origin}

First step

1
The equation r=3r = 3 means every point is at distance 33 from the origin, regardless of θ\theta.

Full solution

  1. 2
    This is the set of all points satisfying x2+y2=9x^2 + y^2 = 9 in rectangular form.
  2. 3
    Therefore, the graph is a circle of radius 33 centered at the origin.
In polar coordinates, r=cr = c (a constant) always gives a circle centered at the origin with radius ∣c∣|c|. This is one of the simplest polar graphs and demonstrates how polar coordinates can express circles very naturally.

Example 2

medium
Identify the type of polar curve r=2+2cos⁡θr = 2 + 2\cos\theta and find key features.

Example 3

medium
Identify the curve r=4+2cos⁡θr = 4 + 2\cos\theta as cardioid, dimpled limaçon, convex limaçon, or inner-loop limaçon.

Example 4

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 5

hard
Find the area inside both circles r=2sin⁡θr = 2\sin\theta and r=2cos⁡θr = 2\cos\theta.

Example 6

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 7

challenge
Find the total area enclosed by both loops of the lemniscate r2=4cos⁡(2θ)r^2 = 4\cos(2\theta).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
How many petals does the rose curve r=3sin⁡(4θ)r = 3\sin(4\theta) have?

Example 2

hard
Find the area enclosed by one petal of the rose curve r=4cos⁡(3θ)r = 4\cos(3\theta).

Example 3

easy
What curve is r=5r = 5?

Example 4

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

Example 5

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

Example 6

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

Example 7

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

Example 8

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

Example 9

easy
Is r=2+3cos⁡θr = 2 + 3\cos\theta a cardioid or a limaçon with an inner loop?

Example 10

easy
Over what interval does r=sin⁡(3θ)r=\sin(3\theta) complete its full graph?

Example 11

medium
Find the maximum rr value of r=2+2sin⁡θr = 2 + 2\sin\theta and the angle where it occurs.

Example 12

medium
At what angles does r=2cos⁡θr = 2\cos\theta pass through the origin?

Example 13

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

Example 14

medium
Identify r=4sin⁡θr = 4\sin\theta as a Cartesian curve.

Example 15

medium
Does r=1+2cos⁡θr = 1 + 2\cos\theta produce negative rr values? Where?

Example 16

medium
What is the difference in shape between r=3+cos⁡θr=3+\cos\theta and r=1+3cos⁡θr=1+3\cos\theta?

Example 17

medium
Find the tip of the petal of r=5cos⁡(2θ)r = 5\cos(2\theta) nearest θ=0\theta = 0.

Example 18

challenge
Sketch-classify r2=9cos⁡(2θ)r^2 = 9\cos(2\theta) and state where it is undefined.

Example 19

challenge
Find all intersection points of r=1r = 1 and r=2cos⁡θr = 2\cos\theta.

Example 20

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 21

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

Example 22

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

Example 23

easy
Identify the type of polar curve given by r=4+4sin⁡θr = 4 + 4\sin\theta.

Example 24

easy
How many petals does the rose curve r=7sin⁡(5θ)r = 7\sin(5\theta) have?

Example 25

easy
How many petals does the rose curve r=2cos⁡(6θ)r = 2\cos(6\theta) have?

Example 26

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

Example 27

easy
What is the smallest non-negative angle at which r=6cos⁡θr = 6\cos\theta achieves its maximum?

Example 28

medium
Find the area enclosed by the full cardioid r=1+cos⁡θr = 1 + \cos\theta.

Example 29

medium
Find the area of one petal of r=2sin⁡(2θ)r = 2\sin(2\theta).

Example 30

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

Example 31

medium
Find the points on r=2+2sin⁡θr = 2 + 2\sin\theta where the graph crosses the pole (origin).

Example 32

medium
Find the maximum distance from the origin on the curve r=4−3cos⁡θr = 4 - 3\cos\theta.

Example 33

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

Example 34

medium
How many petals does r=3sin⁡(7θ)r = 3\sin(7\theta) have?

Example 35

hard
Find the area inside the inner loop of the limaçon r=1+2cos⁡θr = 1 + 2\cos\theta.

Example 36

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

Example 37

hard
Determine the symmetry of r2=4sin⁡(2θ)r^2 = 4\sin(2\theta).

Example 38

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

Example 39

challenge
Find the area inside the cardioid r=1+cos⁡θr = 1 + \cos\theta but outside the circle r=1r = 1.
← Back to Polar Graphs Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

polar coordinatestrigonometric functions