Spherical Coordinates Calculator

Spherical coordinates describe a point in three-dimensional space using one distance and two angles instead of three perpendicular distances. This calculator converts Cartesian coordinates x, y, and z into radius, azimuth, and polar angle. The convention used here measures azimuth counterclockwise in the xy-plane from the positive x-axis and measures the polar angle downward from the positive z-axis. Stating the convention is important because some subjects use elevation in place of polar angle. The radial distance is the straight-line distance from the origin to the point. It is calculated as the square root of x squared plus y squared plus z squared, which is the three-dimensional extension of the Pythagorean theorem. Radius is always nonnegative. At the origin the radius is zero, but both direction angles are undefined, so the calculator asks for a point away from the origin. Azimuth is found with the two-argument arctangent of y and x. Unlike an ordinary arctangent of y divided by x, this operation identifies the correct quadrant and also works when x is zero. The result is normalized to a range from zero degrees up to, but not including, 360 degrees. A point along positive x has azimuth zero, positive y has 90 degrees, negative x has 180 degrees, and negative y has 270 degrees. The polar angle is the inverse cosine of z divided by radius. It ranges from zero degrees at the positive z-axis to 180 degrees at the negative z-axis, with 90 degrees representing the xy-plane. If your application uses elevation measured upward from the xy-plane, subtract the polar angle from 90 degrees. Mathematics, physics, geography, robotics, and computer graphics may label these angles differently, so verify the expected convention before transferring values. All angles are displayed in degrees and rounded to six decimal places, while calculations use full floating-point precision. Negative Cartesian coordinates are supported and naturally select the appropriate direction. Spherical coordinates are especially helpful when a problem has radial symmetry, such as fields around a point source, planetary positions, camera orientation, or integration over a sphere. They can turn complicated Cartesian relationships into simpler expressions organized around distance and direction.