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.


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


Find the X-Coordinate


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


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°


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


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


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


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.


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.

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.