The Lesson

The polar coordinates of a point can be read off from a graph.

How to Read Off the Polar Coordinates a Point

Reading off the polar coordinates of a point is easy.

Question

What are the polar coordinates of the point shown on the graph below.

Step-by-Step:

1

Draw a line from the pole to the point.

Don't forget: In polar coordinates, there is a reference point (called the pole) and a reference direction shown by the horizontal polar axis.

2

Find the length of this line.
  • Use a ruler with the same scale as the polar axis:

  • Use a compass. With the compass needle on the pole, draw an arc through the point to the polar axis:

Either way, the length of the line is 5. 5 is the radial coordinate of the point.

3

Measure the angle of the line from the polar axis (in the counter-clockwise direction).

The angle is 45°. 45° is the angular coordinate of the point.

4

Write down the polar coordinates as a pair of numbers in brackets, separated by a comma. The radial coordinate (5) found in Step 2 goes on the left. The angular coordinate (45°) found in Step 3 goes on the right.

Answer:

Reading Off Point from Polar Coordinates Using the Polar Grid

A polar grid helps us draw a point from 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 points with the same angular coordinate (because all points on the line are the same angle from the polar axis).
To read off the point on the polar grid, see that it lies on the circle with a radius of 5 and on the line at an angle of 45°.

We can read off the polar coordinates as (5, 45°).

Lesson Slides

The slider below gives another example of how to read off the polar coordinates of a point on a graph.

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: