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Converting from Polar to Cartesian Coordinates

(KS3, Year 7)

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

In these formulas:

x is the x-coordinate of the point in Cartesian coordinates.
y is the y-coordinate of the point in Cartesian coordinates.
r is the radial coordinate of the point in polar coordinates.
θ is the angular coordinate of the point in polar coordinates.

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

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. convert_polar_to_cartesian_answer

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: polar_right_angled_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

This page was written by Stephen Clarke.

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