# The Relationship between Cartesian and Polar Co-ordinates

Cartesian co-ordinates can be converted to polar co-ordinates using the following relationships:

where the co-ordinates are defined in the graph below:

# How to Convert from Cartesian to Polar Co-ordinates

Question: What is a point described by the Cartesian co-ordinates (x,y) in polar co-ordinates (r,θ)?

Step 1
Multiply the x co-ordinate by itself.
x × x = x2.

Step 2
Multiply the y co-ordinate by itself.
y × y = y2.

Step 3
Add the results from Step 1 and Step 2 together.
x2 + y2.

Step 4
√x2 + y2.

Find the angle

Step 5
Divide the y co-ordinate by the x co-ordinate.
y/x.

Step 6
Find the inverse tan of the angle.
tan-1 (y/x).

This is the angle θ.

Find the polar co-ordinates

Step 7
In brackets, write the radius, then the angle, separated by a comma.
(r,θ).

# A Real Example of How to Convert from Cartesian to polar Co-ordinates

Question: What is a point described by the Cartesian co-ordinates (3,4) in polar co-ordinates?

Step 1
Multiply the x co-ordinate by itself.
3 × 3 = 92.

Step 2
Multiply the y co-ordinate by itself.
4 × 16 = y2.

Step 3
Add the results from Step 1 and Step 2 together.
9 + 16 = 25.

Step 4
√25 = 5.

Find the angle

Step 5
Divide the y co-ordinate by the x co-ordinate.
4 ÷ 3 = 1.33.

Step 6
Find the inverse tan of the angle.
tan-1 (1.33) = 53°.

This is the angle θ.

Find the polar co-ordinates

Step 7
In brackets, write the radius, then the angle, separated by a comma.
(5,53°).

(5,53°) is the Cartesian co-ordinate (3,4) converted to polar co-ordinates.

# Another Real Example of How to Convert from Cartesian to Polar Co-ordinates

The slider below shows another real example of how to convert from Cartesian to polar co-ordinates.
show

##### Note
WHERE DO THE RELATIONSHIPS BETWEEN x, y, r AND θ COME FROM?

Polar co-ordinates form a right angled triangle:

The radius is the hypotenuse and the angle is... the angle!

The x co-ordinate is the adjacent of the triangle and the y co-ordinate is the opposite of the triangle.

Using Pythagoras' Theorem, the square of the hypotenuse is the sum of the squares of the other two sides.

r2 = x2 + y2

Taking the square root of both sides gives the relationship between r, x and y:

r = √x2 + y2

When the opposite and adjacent are known, use the tangent to find the angle:

θ = tan-1 (y/x).

# SQUARE ROOTS

Finding the radius r requires calculating a square root.

Apart from the square roots of square numbers, most square roots are not nice, rounded numbers.

Sometimes it is more exact to just write a number a square number rather than calculating and rounding it.

For example, the square root of 8 can be written as:

2.8 or √8