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Polar Coordinates Calculator

Make conversions between rectangular (Cartesian) and polar plane coordinates.

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Polar Coordinates Calculator

This calculator is used to convert rectangular to polar coordinates and vice versa for a 2D system. It shows the step-by-step calculation for both conversions.

What are Cartesian and Polar Coordinates?

Cartesian and Polar Coordinates

1.1 Polar Coordinates:

Polar coordinates represent a point that is positioned with respect to the origin (0,0) and an angle from the origin in the reference direction. They are written as (r, θ), such that:

r is the radial distance from the origin to point
θ is the angle between the x-axis and the line joining point with the origin.

1.2 Cartesian (Rectangular) Coordinates:

Cartesian coordinates indicate a point in an XY-plane by a pair of numerical values. These coordinates are signed distances from a point to two perpendicular directed lines. They are usually written as (x, y), such that:

X represents the horizontal distance from the origin (positive to the right, negative to the left).
Y represents the vertical distance from the origin (positive upwards, negative downwards

How to Make Conversion Between Cartesian and Polar Coordinates?

There are two methods of converting between the coordinates:

2.1 Manual Calculations:

2.1.1 To Convert from Cartesian to Polar

For rectangular coordinates (x, y) given in the above picture, you can find polar coordinates (r, θ) as follows:

r = √(x² + y²)

θ = arctan(y / x)

Cartesian to Polar Conversion

2.1.2 To Convert from Polar to Cartesian

Use the following polar equations to cartesian equations for converting:

Polar to Cartesian Conversion

2.2 Using Polar Coordinates Calculator

Select the conversion type
Enter the required coordinate values
Click ‘Calculate’ and get converted form

Cartesian & Polar Coordinates (Solved Examples)

Example 01:

cartesian to polar coordinates

Solution:

Data Given:

Vertical coordinate = y = 2
Horizontal coordinate = x = 6

Calculations:

Step 1: Determine ‘r’

r = √(x² + y²)

r = √(6² + 2²)

r = √(36 + 4)

r = √(40)

r = 6.324555320336759

Step 2: Determine ‘θ’

θ = arctan (y/x

θ = arctan (2/6

θ = arctan (1/3

Example 2

polar to cartesian coordinates

Solution:

Data Given:

r = 2
θ = 45°

Calculations:

Step 1: Determine Horizontal Rectangular Component ‘x’

x = r cos θ

x = 2 ∗ 0.707

x = 1.414

Step 2: Determine Vertical Rectangular Component ‘y’

y = r sin θ

y = 2 * sin(45°)

Sample Cartesian ↔ Polar Conversions

Cartesian (x, y)	Polar (r, θ)
(3, 4)	5, arctan(4/3) ≈ 53.13°
(-2, 1)	√5, arctan(1 / -2) ≈ -26.57°
(0, 5)	5, 90°
(-4, 4)	√32, arctan(4 / -4) ≈ 135°

This polar coordinates calculator can handle all the conversions between coordinates including all ones listed in the above table, with the complete solution shown.