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 r from the origin and an angle \theta from the positive x-axis, written as (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: Polar coordinates describe position by distance and angle rather than horizontal and vertical displacement. They are ideal for problems with circular or rotational symmetry.

Common stuck point: The conversion \theta = \arctan(y/x) only gives the correct angle in Quadrants I and IV. For Quadrants II and III, you must add \pi (or check the signs of x and y separately) to get the right angle.

Sense of Study hint: Plot the Cartesian point first, note which quadrant it is in, then compute r and theta. Check the quadrant matches before finalizing theta.

Worked Examples

Example 1

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

Solution

  1. 1
    Use the conversion formulas: x = r\cos\theta and y = r\sin\theta.
  2. 2
    x = 4\cos\left(\frac{\pi}{3}\right) = 4 \cdot \frac{1}{2} = 2.
  3. 3
    y = 4\sin\left(\frac{\pi}{3}\right) = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}.

Answer

(2, 2\sqrt{3})
Polar coordinates (r, \theta) locate a point by its distance from the origin and angle from the positive x-axis. Converting to rectangular uses x = r\cos\theta and y = r\sin\theta, which come from the right triangle formed by the point, the origin, and the projection onto the x-axis.

Example 2

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

Practice Problems

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

Example 1

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

Example 2

hard
Find all polar representations of the point with rectangular coordinates (0, -5) where -2\pi \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