Polar to Rectangular Coordinates Calculator

Polar and rectangular coordinate systems

A point in the plane can be described in two standard ways. In the rectangular (Cartesian) system, the point is located by its signed distances along two perpendicular axes: x runs horizontally and y runs vertically. In the polar system, the same point is described by its radial distance r from the origin and the angle theta it makes with the positive x-axis, measured counter-clockwise. Both descriptions are complete and exact - they just put different information first. Physics and engineering often favour polar coordinates for problems with rotational symmetry (circular motion, antenna patterns, AC circuits), while rectangular coordinates dominate linear algebra, computer graphics, and most school mathematics.

How the conversion formulas work

The connection between the two systems comes directly from the unit-circle definitions of sine and cosine. If you draw the right triangle from the origin to the point, the hypotenuse is r, the horizontal leg is x, and the vertical leg is y. Basic trigonometry then gives: x = r cos(theta) and y = r sin(theta). Going the other way, the Pythagorean theorem gives r = sqrt(x^2 + y^2), and the angle is recovered with the two-argument arctangent: theta = atan2(y, x). Using atan2 rather than a plain arctan is important because atan2 preserves the correct quadrant for all four quadrant combinations, while arctan(y/x) is ambiguous between Quadrant I and Quadrant III (and between II and IV). The result from atan2 lies in the range (-pi, pi] in radians, or (-180, 180] in degrees. Adding 2*pi (or 360 deg) to any negative result gives the equivalent angle in the range [0, 2*pi).

Negative radius and principal angles

Unlike Cartesian x and y, the radial coordinate r can be negative. A negative r means the point is in the opposite direction: (r, theta) and (-r, theta + pi) represent the same physical location. Most textbooks define r as non-negative and restrict theta to [0, 2*pi), but some contexts - particularly complex numbers written in polar form - allow r < 0. When converting to rectangular coordinates a negative r causes no problem: the cosine and sine formulas handle it automatically. Going from rectangular back to polar, r = sqrt(x^2 + y^2) is always non-negative, and theta = atan2(y, x) gives the principal angle. If you need theta expressed in [0, 360 deg), add 360 to any negative result from atan2.

Practical applications

Polar-to-rectangular conversion comes up in many applied settings. In electrical engineering, phasors in AC circuit analysis are written in polar form (magnitude and phase angle) and must be converted to rectangular form (real and imaginary parts) before addition or subtraction. In robotics and navigation, sensor readings often arrive as a range r and bearing theta, which must be converted to x-y displacements for path planning. Radar and sonar echo plots are naturally polar; converting to rectangular lets them be overlaid on a standard map grid. In computer graphics, polar parametric curves (rose curves, spirals, limacons) are sampled by sweeping theta and converting each (r, theta) to screen pixel (x, y). Scientific notation in complex number arithmetic is essentially polar form, and every complex multiplication is a polar computation.