Converting from Polar to Cartesian Coordinates
(KS3, Year 7)

The Lesson

Polar coordinates can be converted to Cartesian coordinates using the following formulas:

In these formulas: The graph below shows what we mean by the same point defined in polar coordinates (r, θ) and Cartesian coordinates (x, y):

How to Convert from Polar to Cartesian Coordinates

Converting from the polar to the Cartesian coordinates of a point is easy.

Question

What is a point described by the polar coordinates (8, 30°) in Cartesian coordinates?

Step-by-Step:

Find the X-Coordinate

1

Start with the formula:
x-coordinate = r cos θ
Note: cos θ is the cosine of the angle.

2

Find r and θ from the polar coordinates given in the question. In our example, the polar coordinates of the point is (8, 30°). They are represented in the formula by (r, θ).
(r, θ) = (8, 30°) ∴ r = 8, θ = 30°

3

Substitute r and θ into the formula.
x-coordinate = 8 cos (30°) x-coordinate = 8 × 0.87 x-coordinate = 6.9
The x-coordinate is 6.9

Find the Y-Coordinate

4

Start with the formula:
y-coordinate = r sin θ
Note: sin θ is the sine of the angle.

5

Substitute r and θ into the formula.
y-coordinate = 8 sin (30°) y-coordinate = 8 × 0.5 y-coordinate = 4
The y-coordinate is 4

6

Write down the Cartesian coordinates as a pair of numbers in brackets, separated by a comma. The x-coordinate (6.9) found in Step 3 goes on the left. The y-coordinate (4) found in Step 5 goes on the right.

Answer:

The polar coordinates (8, 30°) become (6.9, 4) when converted to Cartesian coordinates.

Lesson Slides

The slider below gives another example of how to convert from polar to Cartesian coordinates. Open the slider in a new tab

Interactive Widget

Here is an interactive widget to help you learn about converting between Cartesian and polar coordinates.

Why Do the Formulas Work?

Polar coordinates form a right triangle:

The radial coordinate is the hypotenuse and the angular coordinate is the angle.