Polar to Cartesian Coordinate Converter

Two Coordinate Systems Cartesian coordinates locate a point using horizontal (x) and vertical (y) distances from the origin. Polar coordinates locate the same point using distance from the origin (r) and angle from the positive x-axis (theta). Both systems describe the same plane, but some problems are much easier in one system than the other.

Polar to Cartesian x = r * cos(theta). y = r * sin(theta). For example, (r=5, theta=30 degrees) gives x = 5cos(30) = 4.33 and y = 5sin(30) = 2.5.

Cartesian to Polar r = sqrt(x^2 + y^2). theta = atan2(y, x). The atan2 function correctly handles all four quadrants, unlike basic arctan which only covers two.

Radians vs Degrees Radians are the natural unit for angles in mathematics: a full circle is 2*pi radians (approximately 6.2832). Degrees are the everyday unit: a full circle is 360 degrees. To convert: radians = degrees * pi / 180. Common angles: 30 degrees = pi/6, 45 degrees = pi/4, 60 degrees = pi/3, 90 degrees = pi/2.

When to Use Each System Polar coordinates simplify problems involving circles, spirals, and rotational symmetry. Cartesian coordinates are better for lines, rectangles, and translations. Many physics problems (planetary orbits, electromagnetic fields, wave functions) are naturally expressed in polar or cylindrical coordinates.

One Point, Infinitely Many Polar Addresses This is the part that catches people out. A Cartesian point has exactly one (x, y). A polar point does not have exactly one (r, theta). Add 360 degrees to the angle and you land on the same spot, so (3, 40 degrees) and (3, 400 degrees) and (3, -320 degrees) are all the same point. Worse, a negative r is legal: it means walk backwards along the ray, so (-3, 40 degrees) is the same point as (3, 220 degrees). The calculator accepts all of these and tells you where the point actually lands.

Going the other way, the answer is forced to be unique, which is why this direction is the safer one. The convention here is r greater than or equal to 0 and theta in the range 0 to 360 degrees, sometimes called the principal value.

Worked Example: Both Directions Start with the polar point (r = 5, theta = 53.13 degrees). x = 5 * cos(53.13) = 3.000, y = 5 * sin(53.13) = 4.000, so the Cartesian point is (3, 4). Now convert back. r = sqrt(9 + 16) = sqrt(25) = 5, and theta = atan2(4, 3) = 53.13 degrees. You get exactly what you started with, which is the check worth doing whenever a conversion feels wrong.

Note that arctan(4/3) also gives 53.13 degrees, and so does arctan(-4/-3). The point (-3, -4) sits in quadrant III at 233.13 degrees, but plain arctan reports 53.13 for it. That single failure is why atan2 exists.