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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:
 x is the xcoordinate of the point in Cartesian coordinates.
 y is the ycoordinate 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.
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?StepbyStep:
Find the XCoordinate
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.
xcoordinate = 8 cos (30°)
xcoordinate = 8 × 0.87
xcoordinate = 6.9
The xcoordinate is 6.9
Find the YCoordinate
4
Start with the formula:
ycoordinate = r sin θ
Note: sin θ is the sine of the angle.
5
Substitute r and θ into the formula.
ycoordinate = 8 sin (30°)
ycoordinate = 8 × 0.5
ycoordinate = 4
The ycoordinate is 4
6
Write down the Cartesian coordinates as a pair of numbers in brackets, separated by a comma.
The xcoordinate (6.9) found in Step 3 goes on the left.
The ycoordinate (4) found in Step 5 goes on the right.
Answer:
The polar coordinates (8, 30°) become (6.9, 4) when converted 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.

The xcoordinate 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 ycoordinate 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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What are Cartesian coordinates? What is the xcoordinate? What is the ycoordinate? 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)