Polar Coordinates

Polar coordinates are used to describe the position of a point on a graph. The image below shows a graph. A point is plotted on the graph (a blue cross) with its polar coordinates written beside it: (5, 45°).

• The pole is a reference point (like an origin).
• The horizontal axis is called the polar axis.

How Do Polar Coordinates Work?

Polar coordinates work by measuring how far the point is from a reference point (called the pole) and what angle it is from a reference direction (called the polar axis).

• First you measure how far the point is from the pole. This is called the radial coordinate.
  
  The point above is 5 units from the pole so its radial coordinate is 5.
• Then you measure the angle of the point (in the counter-clockwise direction) from the polar axis. This is called the angular coordinate.
  
  The point above is 45° from the polar axis so its angular coordinate is 45°.
• The radial coordinate (5) and the angular coordinate (45°) are then written in brackets, separated by a comma. The radial coordinate is on the left, the angular coordinate is on the right.
  
  

Polar Coordinates in General

In general, we write polar coordinates as: r is the radial coordinate. θ is the angular coordinate. r can taken any positive number. θ can be measured in degrees or in radians and is often limited to a whole rotation (0 to 360° or 0 to 2π radians).

The Polar Grid

A polar grid helps us use polar coordinates.

• The concentric circles show us points with the same radial coordinate (because the circles have the same radius).
• The straight lines show us point with the same angular coordinate (because all points on the line are the same angle from the polar axis).

What's in a Name?

Polar coordinates are named because Jacob Bernoulli called the point from which other points are measured the pole and the horizontal line which passes through it the polar axis. The radial coordinate is sometimes called the radius. The angular coordinate is sometimes called the polar angle or the azimuth.

Why Are Polar Coordinates Useful?

Polar coordinates are useful when dealing with circular geometry. All the points 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:

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