Polar Coordinates Examples in Math

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

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

Concept 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.

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: A point is (r,θ)(r,\theta): how far from the origin and at what angle, like radar.

Common stuck point: The procedure for polar coordinates is the easy part; the trap is reading (r,θ)(r,\theta) as (x,y)(x,y). Asking "Is the location given as a distance from the origin plus an angle, rather than horizontal and vertical amounts?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the location given as a distance from the origin plus an angle, rather than horizontal and vertical amounts?

Worked Examples

Example 1

easy
Convert the polar coordinates (4,π3)(4, \frac{\pi}{3}) to rectangular (Cartesian) coordinates.

Answer

(2,23)(2, 2\sqrt{3})

First step

1
Use the conversion formulas: x=rcos⁡θx = r\cos\theta and y=rsin⁡θy = r\sin\theta.

Full solution

  1. 2
    x=4cos⁡(π3)=4⋅12=2x = 4\cos\left(\frac{\pi}{3}\right) = 4 \cdot \frac{1}{2} = 2.
  2. 3
    y=4sin⁡(π3)=4⋅32=23y = 4\sin\left(\frac{\pi}{3}\right) = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}.
Polar coordinates (r,θ)(r, \theta) locate a point by its distance from the origin and angle from the positive xx-axis. Converting to rectangular uses x=rcos⁡θx = r\cos\theta and y=rsin⁡θy = r\sin\theta, which come from the right triangle formed by the point, the origin, and the projection onto the xx-axis.

Example 2

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

Example 3

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 4

medium
Convert the polar equation r=2sec⁡θr = 2\sec\theta to a rectangular equation.

Example 5

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

Example 6

hard
Find all polar representations of the rectangular point (−23,2)(-2\sqrt{3}, 2) with r>0r > 0 and −2π≤θ<2π-2\pi \le \theta < 2\pi.

Example 7

hard
Find the polar equation of the line y=2x+5y = 2x + 5.

Example 8

challenge
Convert the polar equation r=62−cos⁡θr = \dfrac{6}{2 - \cos\theta} to rectangular form and identify the conic.

Practice Problems

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

Example 1

medium
Convert the equation x2+y2=6xx^2 + y^2 = 6x to polar form.

Example 2

hard
Find all polar representations of the point with rectangular coordinates (0,−5)(0, -5) where −2π≤θ<2π-2\pi \le \theta < 2\pi.

Example 3

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

Example 4

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

Example 5

easy
Convert (r,θ)=(6,π)(r,\theta)=(6,\pi) to Cartesian.

Example 6

easy
Find rr for the Cartesian point (3,4)(3,4).

Example 7

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

Example 8

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

Example 9

easy
What is the order of polar coordinates: distance or angle first?

Example 10

easy
Convert the Cartesian point (0,5)(0,5) to polar with r>0r>0.

Example 11

medium
Convert the Cartesian point (1,1)(1,1) to polar with r>0r>0, 0≤θ<2π0\le\theta<2\pi.

Example 12

medium
Convert (−1,3)(-1,\sqrt{3}) to polar with r>0r>0, 0≤θ<2π0\le\theta<2\pi.

Example 13

medium
Convert (r,θ)=(−3,π4)(r,\theta)=(-3,\frac{\pi}{4}) to Cartesian.

Example 14

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

Example 15

medium
Convert the polar equation r=4cos⁡θr=4\cos\theta to Cartesian.

Example 16

medium
Convert x2+y2=9x^2+y^2=9 to a polar equation.

Example 17

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

Example 18

challenge
Convert r=21+cos⁡θr = \frac{2}{1+\cos\theta} to Cartesian and identify the conic.

Example 19

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

Example 20

challenge
A point has polar coordinates (4,5π6)(4,\frac{5\pi}{6}). Find its exact Cartesian coordinates.

Example 21

medium
Convert (0,−4)(0,-4) to polar with r>0r>0, 0≤θ<2π0\le\theta<2\pi.

Example 22

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

Example 23

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

Example 24

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

Example 25

easy
Convert the polar point (r,θ)=(1,7π4)(r,\theta) = (1, \frac{7\pi}{4}) to rectangular coordinates.

Example 26

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

Example 27

medium
Convert the rectangular point (1,−3)(1, -\sqrt{3}) to polar coordinates with r>0r > 0 and 0≤θ<2π0 \le \theta < 2\pi.

Example 28

medium
Express the equation x2+y2=25x^2 + y^2 = 25 in polar form.

Example 29

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

Example 30

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

Example 31

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

Example 32

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

Example 33

medium
Convert the polar equation r2=4cos⁡(2θ)r^2 = 4\cos(2\theta) to a rectangular equation.

Example 34

medium
Convert the polar equation r=41−cos⁡θr = \frac{4}{1 - \cos\theta} to a rectangular equation.

Example 35

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

Example 36

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 37

hard
A point in rectangular coordinates is (x,y)=(−4,43)(x, y) = (-4, 4\sqrt{3}). Write its polar form with r<0r < 0 and 0≤θ<2π0 \le \theta < 2\pi.

Example 38

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.
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Background Knowledge

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

trigonometric functionsunit circleradian measure