Rectangular to Polar Calculator

This rectangular to polar calculator converts a point given as rectangular (Cartesian) coordinates x and y into polar coordinates, a radius r and an angle theta. The radius is the straight line distance from the origin, found with Pythagoras as the square root of x squared plus y squared. The angle is measured anticlockwise from the positive x axis and is found with the two argument arctangent, atan2, which returns the correct angle in all four quadrants. The calculator gives the angle in both degrees and radians. Polar coordinates are useful in physics, engineering, navigation, complex numbers and computer graphics, wherever distance and direction are more natural than horizontal and vertical components. Enter x and y and read the radius and angle. The formula, a worked example and the assumptions are below.

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r = 5
Angle (degrees)53.130102 deg
Angle (radians)0.927295 rad

The formula

For a point (x, y): the radius is r = sqrt(x^2 + y^2), and the angle is theta = atan2(y, x), measured from the positive x axis. Using atan2 rather than plain arctan gives the correct quadrant for the angle.

Worked example

The point (3, 4) has r = sqrt(9 + 16) = 5 and theta = atan2(4, 3), about 53.13 degrees. Enter 3 and 4 above to confirm.

Frequently asked questions

How do I convert (x, y) to polar?

Radius r = sqrt(x^2 + y^2), angle theta = atan2(y, x). The angle is measured anticlockwise from the positive x axis.

Why use atan2 instead of arctan?

Plain arctan(y/x) loses the quadrant and fails when x is zero. atan2(y, x) returns the correct angle for any point.

What range is the angle in?

atan2 returns an angle between -180 and 180 degrees (-pi and pi radians).

Who this calculator is for

This calculator is for students, engineers and anyone converting between Cartesian and polar coordinates.

What this calculator assumes

  • You enter the x and y coordinates.
  • The angle is measured from the positive x axis using atan2.
  • Results are rounded for display.

Formula and sources

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