The Lesson
Cartesian coordinates can be converted to polar coordinates using the following formulas:In these formulas:
- r is the radial coordinate of the point in polar coordinates.
- θ is the angular coordinate of the point in polar coordinates.
- x is the x-coordinate of the point in Cartesian coordinates.
- y is the y-coordinate of the point in Cartesian coordinates.
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.Interactive Widget
Here is an interactive widget to help you learn about converting between Cartesian and polar coordinates.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