Rectangular to Polar Coordinates Calculator

Rectangular coordinates locate a point using horizontal (x) and vertical (y) displacements from an origin. Polar coordinates instead define the exact same point using distance from the origin (radius, r) and an angle of rotation (theta, theta) measured counterclockwise from the positive x-axis. To find the radius, we apply the Pythagorean theorem: r = sqrt(x^2 + y^2). To find the angle, we use the two-argument arctangent function: theta = atan2(y, x), which correctly determines the quadrant of the given coordinate pair.

Let us convert the rectangular coordinate point (3, 4) into polar coordinates. First, calculate the radius by squaring both components: x^2 = 3^2 = 9 and y^2 = 4^2 = 16. Sum these values to get 25, and take the square root: r = sqrt(25) = 5. Next, determine the angle by evaluating theta = atan2(4, 3), which yields approximately 0.927 radians or about 53.13 degrees. Thus, the polar coordinate representation is (5, 53.13°).

When working with angle outputs, always confirm whether your calculator or application is set to degrees or radians to avoid costly calculation errors. Remember that the atan2 function is superior to a standard arctangent because it evaluates the signs of both x and y to place the resultant angle in the correct geometric quadrant automatically.