Polar ↔ Rectangular Converter
Convert between polar (r, θ) and rectangular (x, y) with the formulas and a worked example.
Polar ↔ Rectangular Converter
Convert between polar and rectangular coordinate systems with step-by-step solutions
Coordinate Input
Quick Examples
Coordinate Visualization
Coordinate Systems
Cartesian coordinates specify position using perpendicular x and y distances from origin.
Polar coordinates specify position using distance from origin (r) and angle from positive x-axis (θ).
Two ways to name a point
Rectangular (Cartesian) uses perpendicular distances: (x, y). Polar uses distance from the origin and an angle from the positive x-axis: (r, θ). Same plane; different labels. Grids and screens favor (x, y). Circles, rotations, and bearings favor (r, θ).
Conversion formulas
Rectangular → polar
r = √(x² + y²)
θ = atan2(y, x)
Use atan2 so the quadrant is right. Plain arctan(y/x) is not enough.
Polar → rectangular
x = r cos(θ)
y = r sin(θ)
Match degrees vs radians before calling trig functions.
Angles and gotchas
Full circle: 360° or 2π radians. Degrees feel natural for navigation; radians are what calculus and most math libraries expect. Convert with × π/180 or × 180/π.
- Angles wrap: (5, 30°) and (5, 390°) are the same point
- Origin: r = 0; θ is arbitrary
- Negative r is valid; rewrite as positive r with θ + π if you prefer
- Keep full precision until the last step, then convert back to check
When to pick which
Rectangular
Linear relations, grids, matrices, plots, screen coordinates.
Polar
Rotation, circular motion, direction + range (radar, bearings), complex modulus/argument.
Related mathematical tools
Frequently Asked Questions
When should I use polar?
When distance-from-center and direction are the natural variables—rotation, circular motion, navigation, complex numbers in polar form.
Why do I get different angles for the same point?
Adding or subtracting 360° (2π) does not move the point. Most tools report a principal angle in [0°, 360°) or [0, 2π).
Negative radius?
Valid math. (r, θ) with r < 0 equals (−r, θ + π). Many apps normalize to r ≥ 0.
Degrees vs radians?
Degrees split the circle into 360 parts. One radian is arc length equal to the radius; a full turn is 2π. Mixing them is a common bug.
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