# How to Convert from Cartesian to Polar Co-ordinates (Mathematics Lesson)

# 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)?

**Find the radius**

Step 1

x × x = x

^{2}.

Step 2

y × y = y

^{2}.

Step 3

**Step 1**and

**Step 2**together.

x

^{2}+ y

^{2}.

Step 4

√x

^{2}+ y

^{2}.

This is the radius r.

**Find the angle**

Step 5

y/x.

Step 6

tan

^{-1}(y/x).

This is the angle θ.

**Find the polar co-ordinates**

Step 7

(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?

**Find the radius**

Step 1

3 × 3 = 9

^{2}.

Step 2

4 × 16 = y

^{2}.

Step 3

**Step 1**and

**Step 2**together.

9 + 16 = 25.

Step 4

√25 = 5.

This is the radius r.

**Find the angle**

Step 5

4 ÷ 3 = 1.33.

Step 6

tan

^{-1}(1.33) = 53°.

This is the angle θ.

**Find the polar co-ordinates**

Step 7

(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.##### Interactive Test

**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.

r

^{2}= x

^{2}+ y

^{2}

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

r = √x

^{2}+ y

^{2}

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