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What Are Polar Co-ordinates? (Mathematics Lesson)

What Are Polar Co-ordinates?

Polar co-ordinates can be used to describe a point, called a pole, on a graph.

Polar co-ordinates are described by a pair of numbers in a bracket, separated by a comma.

  • The number on the left gives the radius: how far the pole from the origin.

  • The number on the right gives the angle: the angle of the pole from the horizontal axis.

How to Describe a Point in Cartesian Co-ordinates

Question: What are the polar co-ordinates of the point shown on the graph below?

Step 1
Draw a line from the origin to the pole.

Step 2
Find the length of the line. This is the radius.

Step 3
Find the angle of the line from the horizontal axis. This is the angle.

Step 3
Write the polar co-ordinates as
( radius, angle )

The co-ordinate is (5,45°).

How to Draw a Pole in Polar Co-ordinates

Question: Draw the pole described by the polar co-ordinates (3,30°) on a graph.

Step 1
Draw a line at the angle from the horizontal axis.

Step 2
Along this line, measure a length 3 from the origin. The pole described by the polar co-ordinates is found at the end of this line.

Interactive Widget
Here is an interactive widget to help you learn about Cartesian and polar_coordinates.
Interactive Test

Polar co-ordinates are so named because the points are called poles.

The radius is sometimes called the radial co-ordinate.

The angle is sometimes called the angular co-ordinate, the polar angle or the azimuth.


Polar co-ordinates are useful when dealing with circular geometry.

For instance, all the poles that can be drawn on a circumference of a circle have the same radius, but lie at difference angles.

For example, a circle of radius 2:


An easy mistake to make is to mix up the radius and the angle.

  • The co-ordinate on the left is the radius.

    It describes how far from the origin the pole is.

  • The co-ordinate on the right is the angle.

    It describes the angle of the pole from the horizontal axis.