Cartesian To Polar
Convert any point (x, y) from Cartesian to polar coordinates instantly. Get radial distance r = sqrt(x^2+y^2) and angle theta for any (x, y) input.
About This Calculator
The Cartesian to Polar coordinate converter transforms any point (x, y) in the 2D Cartesian plane into its equivalent polar representation (r, theta). The radial distance r represents the straight-line distance from the origin to the point, calculated as r = sqrt(x^2 + y^2). The angle theta, measured in degrees, represents the direction of the point from the origin, measured counterclockwise from the positive x-axis.
This conversion is fundamental in mathematics, physics, engineering, and computer graphics. Polar coordinates simplify calculations involving circular motion, forces, waves, and fields. For example, in physics, polar coordinates make it easier to describe the motion of planets, pendulums, and rotating systems. The formula uses the atan2(y, x) function which automatically determines the correct quadrant for the angle based on the signs of x and y.
This calculator supports all real-number inputs including positive, negative, zero, and fractional values. Results show r with up to four decimal places of precision and theta in degrees ranging from -180 deg to 180 deg.
Frequently Asked Questions
How do I convert Cartesian coordinates to polar coordinates?
To convert (x, y) to polar (r, theta), compute r = sqrt(x^2 + y^2) for the radial distance and theta = arctan(y / x) using the atan2 function to get the correct quadrant. The result gives both the distance from the origin and the angle measured counterclockwise from the positive x-axis.
What is the formula for converting Cartesian to polar?
The conversion formulas are r = sqrt(x^2 + y^2) for the radius and theta = atan2(y, x) for the angle. The angle theta is given in radians by default but this calculator shows degrees. Multiply radians by 180/pi to get degrees.
What is the range of theta (theta) in polar coordinates?
The angle theta typically ranges from -pi to pi (-180 deg to 180 deg) using the atan2 function, or from 0 to 2pi (0 deg to 360 deg) depending on convention. This calculator returns theta in degrees from -180 deg to 180 deg.
What happens when x = 0 in cartesian to polar conversion?
When x = 0, the point lies on the y-axis. If both x = 0 and y = 0, the result is r = 0 and theta is undefined (the origin). If x = 0 and y > 0, then theta = 90 deg. If x = 0 and y < 0, then theta = -90 deg.
How is polar form different from Cartesian form?
Cartesian coordinates (x, y) describe a point by its horizontal and vertical distances from the origin. Polar coordinates (r, theta) describe the same point by its straight-line distance from the origin (r) and the angle (theta) from the positive x-axis. Both systems can represent any point in a 2D plane.
Can this calculator handle negative coordinates?
Yes, this calculator accepts both positive and negative values for x and y coordinates. The atan2 function correctly determines the angle based on the signs of both coordinates, placing the point in the proper quadrant.
Is this cartesian to polar converter free to use?
Yes, all calculators on Calculy are completely free to use with no registration required. You can also share your results via the share button to get a link with your input values pre-filled.