## The Lesson

Cartesian coordinates can be converted to polar coordinates using the following formulas:

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

## How to Convert from Cartesian to Polar Coordinates

Converting from the Cartesian to the polar coordinates of a point is easy.

## Question

What is a point described by the Cartesian coordinates (3, 4) in polar coordinates?

# 1

$$Radial\:coordinate = \sqrt{x^2 + y^2}$$

# 2

Find x and y from the Cartesian coordinates given in the question. In our example, the Cartesian coordinates of the point is (3, 4). They are represented in the formula by (x, y).
(x, y) = (3, 4) ∴ x = 3, y = 4

# 3

Substitute x and y into the formula.
$$Radial\:coordinate = \sqrt{3^2 + 4^2}$$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = \sqrt{(3 \times 3) + (4 \times 4)}$$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = \sqrt{9 + 16}$$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = \sqrt{25}$$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = 5$$

# 4

$$Angular\:coordinate = tan^{-1} \Big(\frac{y}{x}\Big)$$
Note: tan−1 is the inverse tangent function.

# 5

Substitute x and y into the formula.
$$Angular\:coordinate = tan^{-1} \Big(\frac{4}{3}\Big)$$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = tan^{-1} \Big(1.33\Big)$$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = 53.1°$$
The angular coordinate is 53.1°

# 6

Write down the polar coordinates as a pair of numbers in brackets, separated by a comma. The radial coordinate (5) found in Step 3 goes on the left. The angular coordinate (53.1°) found in Step 5 goes on the right.

The Cartesian coordinates (3, 4) become (5, 53.1°) when converted to polar coordinates.

## Lesson Slides

The slider below gives another example of how to convert from Cartesian to polar coordinates. Open the slider in a new tab

## Why Do the Formulas Work?

Polar coordinates form a right triangle:

The radial coordinate is the hypotenuse and the angular coordinate is the angle. Using Pythagoras' Theorem, the square of the hypotenuse is the sum of the squares of the other two sides. The x-coordinate is the adjacent of the triangle and the y-coordinate is the opposite of the triangle.
$$r^2 = x^2 + y^2$$
Taking the square root of both sides gives the relationship between r, x and y:
$$r = \sqrt{x^2 + y^2}$$
When the opposite and adjacent are known, use the tangent to find the angle:
$$\theta = tan^{-1} \Big(\frac{y}{x}\Big)$$

## Square Roots

Finding the radial coordinate r requires finding a square root. Apart from the square roots of square numbers, most square roots are not whole numbers. Sometimes it is more exact to just write a number as a square number rather than calculating and rounding it. For example, the square root of 8 can be written as:
2.8 or √8