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?

Step-by-Step:

Find the Radial Coordinate

1

Start with the formula:

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

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.

$$Radial\:coordinate = \sqrt{3^2 + 4^2}$$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = \sqrt{(3 \times 3) + (4 \times 4)}$$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = \sqrt{9 + 16}$$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = \sqrt{25}$$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: = 5$$

The radial coordinate is 5

Find the Angular Coordinate

4

Start with the formula:

$$Angular\:coordinate = tan^{-1} \Big(\frac{y}{x}\Big)$$

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.

Answer:

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

Slider

The slider below gives another example of how to convert from Cartesian to polar coordinates.

Open the slider in a new tab

See Also

What are Cartesian coordinates? What is the x-coordinate? What is the y-coordinate? What is a right triangle? What is the hypotenuse? What is the adjacent? What is the opposite? What is Pythagoras' Theorem? What is a square root? Using the tangent function to find the angle Learn more about converting between Cartesian and polar coordinates ( interactive widget)