# How to Convert from Cartesian to Polar Coordinates

## Converting from Cartesian to Polar Coordinates

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.