Graphing Polar Equations: Shapes and Steps

Graphing polar equations means plotting every point $(r, \theta)$ that makes an equation like $r = 2 + 2 \cos \theta$ true. Make a table of $\theta$ and $r$ values, plot the points on a polar grid, and connect them in order. Learn the common shapes: circles, rose curves, cardioids and limaçons.

Before this lesson: You should know Polar Coordinates, The Unit Circle and Graphs of Sine and Cosine.

How Polar Graphs Work

In a polar equation, $r$ depends on $\theta$ . As the angle sweeps around from 0 to $2\pi$ , the distance from the pole grows and shrinks. The curve traces where those points land.

If $r$ comes out negative, plot the point in the opposite direction. For example, $r = -3$ at $\theta = \frac{\pi}{2}$ lands 3 units straight down, not up.

Three polar graphs on grids with circles of radius 2 and 4. Left, solid blue: the rose r = 3 cos 2θ with four petals of length 3 along the axes. Middle, dashed orange: the cardioid r = 2 + 2 cos θ, a heart shape that reaches 4 on the right and touches the pole on the left. Right, dotted green: the limaçon r = 1 + 2 sin θ, reaching 3 at the top with a small inner loop inside it, just above the pole. — Three common polar shapes: a four-petal rose, a cardioid and a limaçon with an inner loop.

Common Polar Graphs

Equation	Shape	Key facts
$r = a$	Circle at the pole	Radius $\lvert a \rvert$
$\theta = \alpha$	Line through the pole	Makes angle $\alpha$ with the polar axis
$r = a \cos \theta$ or $r = a \sin \theta$	Circle through the pole	Diameter $\lvert a \rvert$
$r = a \cos n\theta$ or $r = a \sin n\theta$	Rose curve	$n$ petals if $n$ is odd, $2n$ if $n$ is even; petal length $\lvert a \rvert$
$r = a + a \cos \theta$ (or with $\sin$ , or minus)	Cardioid	Heart shape that touches the pole
$r = a + b \cos \theta$ (or with $\sin$ )	Limaçon	Inner loop if $\lvert a \rvert \lt \lvert b \rvert$ ; dimple if $\lvert b \rvert \lt \lvert a \rvert \lt 2\lvert b \rvert$ ; no dimple if $\lvert a \rvert \ge 2\lvert b \rvert$

Symmetry shortcuts

Equations with $\cos \theta$ are symmetric about the polar axis (the x-axis), because $\cos(-\theta) = \cos \theta$ .
Equations with $\sin \theta$ are symmetric about the line $\theta = \frac{\pi}{2}$ (the y-axis).
Use symmetry to plot half the points, then mirror them.

Steps for Graphing a Polar Equation

Identify the shape from the table above, if you can.
Make a table of values for $\theta$ from 0 to $2\pi$ , using unit circle angles.
Plot each point $(r, \theta)$ . For a negative $r$ , go the opposite way.
Connect the points in order of $\theta$ , and use symmetry to check.

Worked Examples

Example 1: A circle through the pole

Graph $r = 4 \sin \theta$ .

Make a table for $0 \le \theta \le \pi$ : at $\theta = 0$ , $r = 4 \sin 0 = 0$

r = 4 \cdot \frac{1}{2} = 2

At $\theta = \frac{\pi}{6}$ .

r = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3} \approx 3.46

At $\theta = \frac{\pi}{3}$ .

r = 4 \cdot 1 = 4

At $\theta = \frac{\pi}{2}$ .

At $\theta = \frac{5\pi}{6}$ and $\theta = \pi$ , the values repeat in reverse: $r = 2$ , then $r = 0$

Plot and connect: the points form a circle through the pole, reaching up to $(4, \frac{\pi}{2})$

Answer: a circle with diameter 4, centered at $(0, 2)$ in rectangular coordinates. From $\pi$ to $2\pi$ , $r$ is negative and the graph traces the same circle again.

Example 2: A cardioid

Graph $r = 2 + 2 \cos \theta$ .

Identify the shape: $a = b = 2$ , so it is a cardioid, symmetric about the polar axis

r = 2 + 2(1) = 4

At $\theta = 0$ .

r = 2 + 2 \cdot \frac{1}{2} = 3

At $\theta = \frac{\pi}{3}$ .

r = 2 + 2(0) = 2

At $\theta = \frac{\pi}{2}$ .

r = 2 + 2(-\frac{1}{2}) = 1

At $\theta = \frac{2\pi}{3}$ .

At $\theta = \pi$ : $r = 2 + 2(-1) = 0$ , so the curve touches the pole

Mirror these points across the polar axis for $\pi \lt \theta \lt 2\pi$

Answer: a heart shape that reaches 4 on the right and has its point at the pole, as in the middle graph above.

Example 3: A rose curve

Graph $r = 3 \cos 2\theta$ .

Identify the shape: a rose with $n = 2$ . Since $n$ is even, there are $2n = 4$ petals, each 3 long

At $\theta = 0$ : $r = 3 \cos 0 = 3$ , the tip of a petal on the polar axis

r = 3 \cos \frac{\pi}{3} = 1.5

At $\theta = \frac{\pi}{6}$ .

r = 3 \cos \frac{\pi}{2} = 0

At $\theta = \frac{\pi}{4}$ (back at the pole).

At $\theta = \frac{\pi}{2}$ : $r = 3 \cos \pi = -3$ . Plot it 3 units in the opposite direction, straight down

Keep going around: the petals point along $\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$

Answer: a four-petal rose with petal tips 3 units out on each axis, as in the left graph above.

Example 4: A limaçon with an inner loop

Graph $r = 1 + 2 \sin \theta$ .

Identify the shape: $a = 1$ and $b = 2$ , and $1 \lt 2$ , so it is a limaçon with an inner loop, symmetric about $\theta = \frac{\pi}{2}$

Find the largest $r$ : at $\theta = \frac{\pi}{2}$ , $r = 1 + 2(1) = 3$

Find where it passes through the pole: $1 + 2 \sin \theta = 0$ , so $\sin \theta = -\frac{1}{2}$ and $\theta = \frac{7\pi}{6}, \frac{11\pi}{6}$

Between those angles, $r$ is negative. At $\theta = \frac{3\pi}{2}$ : $r = 1 + 2(-1) = -1$ , which plots 1 unit straight up

r = 1 + 2(0) = 1

At $\theta = 0$ and $\theta = \pi$ .

Answer: a large loop reaching 3 at the top, with a small inner loop that reaches 1 above the pole, as in the right graph above.

Example 5: Word problem with a microphone

A cardioid microphone picks up a singer within $r = 5 + 5 \cos \theta$ feet, where $\theta = 0$ points straight out the front. How far away can the singer be when standing in front, to the side ( $\theta = \frac{\pi}{2}$ ), behind, and at $\theta = \frac{\pi}{3}$ ?

r = 5 + 5(1) = 10

In front, $\theta = 0$ (feet).

r = 5 + 5(0) = 5

To the side, $\theta = \frac{\pi}{2}$ (feet).

r = 5 + 5(-1) = 0

Behind, $\theta = \pi$ (feet).

r = 5 + 5 \cdot \frac{1}{2} = 7.5

At $\theta = \frac{\pi}{3}$ (feet).

Answer: 10 ft in front, 5 ft to the side, 7.5 ft at $\frac{\pi}{3}$ , and none from directly behind. That is why these microphones block noise from the back.

Common Mistakes

Mistake	Example of the error	Fix
Plotting negative r on the same ray	Plotting $r = -3$ at $\theta = \frac{\pi}{2}$ as 3 units up	Negative $r$ goes the opposite way: 3 units down.
Wrong number of petals	Saying $r = 3 \cos 2\theta$ has 2 petals	If $n$ is even, there are $2n$ petals: 4.
Too few points	Plotting only $\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$	Use steps of $\frac{\pi}{6}$ or smaller so you see the shape.
Mixing up the limaçon types	Calling $r = 1 + 2 \sin \theta$ a cardioid	A cardioid needs $\lvert a \rvert = \lvert b \rvert$ . Here $1 \lt 2$ , so there is an inner loop.
Stopping at $\pi$	Graphing a cardioid only for $0 \le \theta \le \pi$	Most curves need $0 \le \theta \lt 2\pi$ . Use symmetry to fill in the rest.

Related Methods

Convert to rectangular form

Circles are easy to check by converting. Multiply $r = 4 \sin \theta$ by $r$ to get $x^2 + y^2 = 4y$ , a circle, as shown in polar coordinates.

Use a graphing calculator

Set the calculator to polar mode, enter $r$ as a function of $\theta$ , and set $\theta$ from 0 to $2\pi$ . Compare the screen with your hand-drawn sketch.

Think of it like a sine graph

The table of $r$ values follows the same ups and downs as the sine and cosine graphs. Picture $r$ against $\theta$ first, then wrap that graph around the pole.

Practice Problems

Name the graph of $r = 5$ .
Name the graph of $\theta = \frac{\pi}{3}$ .
Name the graph of $r = 6 \cos \theta$ and give its center and radius in rectangular coordinates.
How many petals does $r = 4 \sin 3\theta$ have, and how long is each?
How many petals does $r = 2 \cos 4\theta$ have?
Name the graph of $r = 3 - 3 \sin \theta$ . Find its largest $r$ and the angle where it happens.
Name the graph of $r = 2 + 4 \cos \theta$ and find where it passes through the pole on $[0, 2\pi)$ .
Name the graph of $r = 4 + 2 \cos \theta$ . Find its largest and smallest $r$ .
For $r = 2 \sin 2\theta$ , find $r$ at $\theta = \frac{\pi}{12}, \frac{\pi}{4}, \frac{5\pi}{12}, \frac{\pi}{2}$ .
A microphone picks up sound within $r = 4 + 4 \cos \theta$ feet. How far away can a singer be at $\theta = \frac{2\pi}{3}$ ? At $\theta = \frac{\pi}{4}$ ?

Show answers

$r$ is always 5, at every angle
Answer: a circle centered at the pole with radius 5

Every point has angle $\frac{\pi}{3}$ , with any $r$ , positive or negative
Answer: a line through the pole at angle $\frac{\pi}{3}$

r^2 = 6r \cos \theta

Multiply by $r$ .

x^2 + y^2 = 6x

Replace r squared with x squared plus y squared, and r cos θ with x.

x^2 - 6x + 9 + y^2 = 9

Complete the square.

(x - 3)^2 + y^2 = 9

Factor the perfect square.

Answer: a circle through the pole, center $(3, 0)$ , radius 3

$n = 3$ is odd, so there are 3 petals
Petal length is $\lvert a \rvert = 4$
Answer: 3 petals, each 4 long

$n = 4$ is even, so there are $2n = 2 \cdot 4 = 8$ petals
Answer: 8 petals, each 2 long

$a = b = 3$ , so it is a cardioid, symmetric about $\theta = \frac{\pi}{2}$

r = 3 - 3(-1) = 6

$r$ is largest when $\sin \theta = -1$ , at $\theta = \frac{3\pi}{2}$ .

Answer: a cardioid with largest $r = 6$ at $\theta = \frac{3\pi}{2}$

$\lvert a \rvert = 2 \lt \lvert b \rvert = 4$ , so it is a limaçon with an inner loop

2 + 4 \cos \theta = 0

Set $r = 0$ .

\cos \theta = -\frac{1}{2}

Subtract 2 and divide by 4.

Answer: an inner-loop limaçon through the pole at $\theta = \frac{2\pi}{3}$ and $\frac{4\pi}{3}$

$\lvert a \rvert = 4 \ge 2\lvert b \rvert = 4$ , so it is a limaçon with no dimple
Largest: $\theta = 0$ gives $r = 4 + 2(1) = 6$
Smallest: $\theta = \pi$ gives $r = 4 + 2(-1) = 2$
Answer: a limaçon with no dimple, $r$ from 2 to 6

$\theta = \frac{\pi}{12}$ : $r = 2 \sin \frac{\pi}{6} = 2 \cdot \frac{1}{2} = 1$
$\theta = \frac{\pi}{4}$ : $r = 2 \sin \frac{\pi}{2} = 2$
$\theta = \frac{5\pi}{12}$ : $r = 2 \sin \frac{5\pi}{6} = 1$
$\theta = \frac{\pi}{2}$ : $r = 2 \sin \pi = 0$
Answer: $r = 1, 2, 1, 0$ , one petal in Quadrant I

10.

r = 4 + 4(-\frac{1}{2}) = 4 - 2 = 2

Substitute $\frac{2\pi}{3}$ . The distance is in feet.

r = 4 + 4 \cdot \frac{\sqrt{2}}{2} = 4 + 2\sqrt{2} \approx 6.83

Substitute $\frac{\pi}{4}$ . The distance is in feet.

Answer: 2 ft and about 6.83 ft

FAQ

How do you graph a polar equation?

Make a table of $\theta$ and $r$ values, plot each point on a polar grid, and connect them in order. Knowing the basic shapes helps you check the result.

How do you know how many petals a rose curve has?

For $r = a \cos n\theta$ or $r = a \sin n\theta$ , there are $n$ petals when $n$ is odd and $2n$ petals when $n$ is even.

What is the difference between a cardioid and a limaçon?

Both have the form $r = a + b \cos \theta$ or $r = a + b \sin \theta$ . A cardioid is the special limaçon with $\lvert a \rvert = \lvert b \rvert$ , so it just touches the pole.

What does a negative r mean on a polar graph?

It means plot the point in the opposite direction from the angle. That is how a rose gets petals on the other side and how a limaçon gets its inner loop.

Summary

Graph by making a table of $\theta$ and $r$ , plotting, and connecting in order.
$r = a$ is a circle; $\theta = \alpha$ is a line; $r = a \cos \theta$ and $r = a \sin \theta$ are circles through the pole.
Roses $r = a \cos n\theta$ : $n$ petals if $n$ is odd, $2n$ if even.
Cardioids have $\lvert a \rvert = \lvert b \rvert$ ; limaçons with $\lvert a \rvert \lt \lvert b \rvert$ have an inner loop.
Cosine graphs are symmetric about the polar axis; sine graphs about $\theta = \frac{\pi}{2}$ .

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