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

homesitemapgraphs and coordinatesconverting 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?

## 1

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

## 4

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.

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.

## Why Do the Formulas Work?

Polar coordinates form a right triangle: The radial coordinate is the hypotenuse and the angular coordinate is the angle.
• The x-coordinate 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-coordinate is the opposite of the triangle. When the hypotenuse and angle are known, use the sine to find the opposite:
y = r sin θ

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