Follow Us on Twitter

How to Convert from Polar to Cartesian Co-ordinates (Mathematics Lesson)

The Relationship between Polar and Cartesian Co-ordinates

Polar co-ordinates can be converted to Cartesian co-ordinates using the following relationships:



where the co-ordinates are defined in the graph below:



How to Convert from Polar to Cartesian Co-ordinates

Question: What is a point described by the polar co-ordinates (r,θ) in Cartesian co-ordinates (x,y)?

Find the x co-ordinate

Step 1
Find the cosine of the angle θ.
cos θ.

Step 2
Multiply by the radius r.
r × cos θ = r cos θ.

This is the x co-ordinate.

Find the y co-ordinate

Step 3
Find the sine of the angle θ.
sin θ.

Step 4
Multiply by the radius r.
r × sin θ = r sin θ.

This is the y co-ordinate.

Find the Cartesian co-ordinates

Step 5
In brackets, write the x co-ordinate, then the y co-ordinate, separated by a comma.
(x,y).

A Real Example of How to Convert from Polar to Cartesian Co-ordinates

Question: What is a point described by the polar co-ordinates (8,30°) in Cartesian co-ordinates?

Find the x co-ordinate

Step 1
Find the cosine of the angle 30°.
cos 30° = 0.87.

Step 2
Multiply by the radius 8.
8 × 0.87 = 7.

This is the x co-ordinate.

Find the y co-ordinate

Step 3
Find the sine of the angle 30°.
sin 30° = 0.5.

Step 4
Multiply by the radius 8.
8 × 0.5 = 4.

This is the y co-ordinate.

Find the Cartesian co-ordinates

Step 5
In brackets, write the x co-ordinate, then the y co-ordinate, separated by a comma.
(7,4).

(7,4) is the polar co-ordinate (8, 30°) converted to Cartesian co-ordinates.

Another Real Example of How to Convert from Polar to Cartesian Co-ordinates

The slider below shows another real example of how to convert from polar to Cartesian co-ordinates.

Have a Go!

Learn more about Cartesian co-ordinates, polar co-ordinates and how to convert between them using this interactive tool.
Interactive Test
  show
 
Note
WHERE DO THE RELATIONSHIPS BETWEEN x, y, r AND θ COME FROM?

Polar co-ordinates form a right angled triangle:



The radius is the hypotenuse and the angle is... the angle!

The x co-ordinate is the adjacent of the triangle.

When the hypotenuse and angle are known, use the cosine to find the adjacent:

x = r cos θ.

The y co-ordinate is the opposite of the triangle.

When the hypotenuse and angle are known, use the sine to find the opposite:

y = r sin θ.