# 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

cos θ.

Step 2

r × cos θ = r cos θ.

This is the x co-ordinate.

**Find the y co-ordinate**

Step 3

sin θ.

Step 4

r × sin θ = r sin θ.

This is the y co-ordinate.

**Find the Cartesian co-ordinates**

Step 5

(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

cos 30° = 0.87.

Step 2

8 × 0.87 = 7.

This is the x co-ordinate.

**Find the y co-ordinate**

Step 3

sin 30° = 0.5.

Step 4

8 × 0.5 = 4.

This is the y co-ordinate.

**Find the Cartesian co-ordinates**

Step 5

(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.