Cartesian coordinates describe the position of a point as a pair (x, y): the horizontal distance from the y-axis and the vertical distance from the x-axis. Polar coordinates describe the same point as a pair (r, θ): r is the straight-line distance of the point from the origin (0,0), and θ (theta) is the angle between the line from the origin to the point and the positive x-axis, usually in degrees or radians.

Use two formulas: r = √(x² + y²) and θ = atan2(y, x). Example: the point (3, 4) gives r = √(9+16) = 5 and θ = atan2(4, 3) = 53.1301° (0.9273 radians), so its polar coordinates are (5; 53.1301°).

The plain arctan(y/x) function only returns angles between −90° and 90°, so it lands in the wrong quadrant for points in quadrants II and III. Example: the point (−5, 12) — arctan(12/−5) = −67.38° (quadrant IV, wrong), while atan2(12, −5) = 112.6199° (quadrant II, correct). The atan2(y, x) function considers the signs of x and y separately, so the angle always lands in the right quadrant.

The angle θ is measured from the positive x-axis, rotating counterclockwise. This calculator displays θ in degrees over the range 0° to 360°, along with its radian equivalent (0 to 2π) and its negative form (−180° to 180°). You can also enter θ in radians when converting polar to Cartesian.

Use x = r × cos(θ) and y = r × sin(θ), with θ in consistent units (degrees or radians). Example: (r=10, θ=30°) gives x = 10 × cos(30°) = 8.6603 and y = 10 × sin(30°) = 5, so the Cartesian coordinates are (8.6603; 5).

At the origin (0,0), the distance r = 0 but the angle θ is mathematically undefined; this calculator uses the common convention θ = 0°. If you enter a negative r, for example (r=−5, θ=30°), it is normalized to r = 5 with the angle shifted by 180°, i.e. (5; 210°), because the vector points the opposite way.

This conversion is used in navigation and radar (range + bearing), robotics (robot arm control), computer graphics and game development (object rotation), physics and engineering (force vector analysis, circular motion), signal processing (phasors), and complex-number mathematics (the a+bi form is equivalent to polar coordinates).

Complex-number-calculator is a complex-number arithmetic calculator (addition, subtraction, multiplication, and division of a+bi forms). This Cartesian to Polar Converter tool specializes in two-way planar coordinate conversion (x,y) ↔ (r,θ) with an interactive vector diagram, degree/radian angle unit options, calculation steps, and worked examples. The two complement each other: the complex number a+bi can indeed be viewed as the point (a,b) on the coordinate plane.