Polar Coordinates Calculator

Cartesian and polar coordinates are two ways to describe the same point in a plane. Cartesian coordinates use a horizontal value x and a vertical value y measured from the origin. Polar coordinates instead use a radius r, the point's straight-line distance from the origin, and an angle theta measured from the positive x-axis. Cartesian notation emphasizes rectangular movement, while polar notation emphasizes distance and direction. To convert a Cartesian point to polar form, the radius follows from the Pythagorean theorem: r is the square root of x squared plus y squared. The angle comes from the two-argument arctangent function, commonly written atan2 of y and x. Unlike a basic inverse tangent, atan2 uses the signs of both coordinates to identify the correct quadrant. This calculator normalizes the result to an angle from zero degrees up to, but not including, 360 degrees. At the origin the direction is conventionally displayed as zero even though every angle reaches the same point. The reverse conversion projects the radius onto the coordinate axes. Multiply r by the cosine of theta to obtain x, and multiply r by the sine of theta to obtain y. The calculator accepts the angle in degrees and converts it internally to radians before using the trigonometric functions. Tiny floating-point remnants near zero are cleaned from the display, so the conversion of radius 10 at 90 degrees appears as x equal to zero rather than an extremely small scientific-notation value. Polar coordinates are especially useful whenever rotation or radial distance is central to a problem. Engineers use them for alternating-current phasors and rotating machinery. Mathematicians use them to graph circles, spirals, and rose curves, or to simplify double integrals over circular regions. Navigators can express a displacement as a bearing and distance, while computer graphics systems use polar relationships for animation, particle effects, and object placement around a center. A single Cartesian point has many equivalent polar descriptions because adding or subtracting a full 360-degree turn leaves the direction unchanged. Negative radii can also be rewritten by reversing direction, but this calculator requires a nonnegative radius to provide one clear conventional output. Use either conversion direction to check trigonometry, translate geometry data, prepare graphing coordinates, or verify calculations involving magnitude and direction.