The Lesson

A point can be drawn on a graph. The polar coordinates of the point tell us where to draw the point on the graph. Imagine you wanted to draw a point with polar coordinates (5, 45°).

How to Draw a Point from Polar Coordinates

Drawing a point from polar coordinates is easy. Polar coordinates have 2 numbers.

  • The number on the left (5) is called the radial coordinate. It tells you how far the point is from a reference point (called the pole).
  • The number on the right (45°) is called the angular coordinate. It tells you what the angle (in the counter-clockwise direction) the point is from a reference direction (called the polar axis).

Question

Draw the point with polar coordinates (5, 45°) on a graph.

Step-by-Step:

1

Start with a graph.

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 first number in the polar coordinates (called the radial coordinate). In our example, the radial coordinate is 5.
(5, 45°)

3

Starting from the pole, count along the polar axis until you find the number found in Step 2 (5).

4

Using a compass, with the compass needle on the pole, draw an arc with a radius given in Step 2 (5).

5

Find the second number in the polar coordinates (called the angular coordinate). In our example, the angular coordinate is 45°.
(5, 45°)

6

Using a protractor, measure the angle found in Step 5 (45°) in the counter-clockwise direction.

7

Draw the point where the line from Step 6 meets the arc drawn in Step 4.

Answer:

Drawing a 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 draw the point (5, 45°) on the polar grid, find the circle with a radius of 5 and the line at an angle of 45°.

We can draw the point straight off, without the use of a compass and protractor:

Lesson Slides

The slider below gives another example of how to draw a point on a graph from the point's polar coordinates.

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: