How to Convert from Polar to Cartesian Coordinates

Converting from Polar to Cartesian Coordinates

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.

Slider

The slider below gives another example of how to convert from polar to Cartesian coordinates.

Open the slider in a new tab

See Also

What are Cartesian coordinates? What is the x-coordinate? What is the y-coordinate? What is the cosine function? What is the sine function? What is a right triangle? What is the hypotenuse? What is the adjacent? What is the opposite? Using the cosine function to find the adjacent Using the sine function to find the opposite Learn more about converting between Cartesian and polar coordinates ( interactive widget)