Polar / Cartesian Converter
Radius (r)5.0000
Angle (degrees)53.1301°
Angle (radians)0.9273
The Polar / Cartesian Converter translates a point between rectangular (x, y) coordinates and polar (r, θ) coordinates in either direction. Cartesian coordinates describe a point by its horizontal and vertical offsets, while polar coordinates describe the same point by its distance from the origin and the angle it makes — two views of the same location that are each more convenient for different problems.
Formula
r = √(x² + y²), θ = atan2(y, x) ; x = r·cos(θ), y = r·sin(θ)
- x
- Horizontal Cartesian coordinate
- y
- Vertical Cartesian coordinate
- r
- Radial distance from the origin
- θ
- Angle measured from the positive x-axis
How it works
- Pick a direction: Cartesian → Polar converts an (x, y) point, while Polar → Cartesian converts a radius and angle back to (x, y).
- Going to polar, the radius is r = √(x² + y²) and the angle is θ = atan2(y, x), which returns the correct quadrant from −180° to 180°.
- Going to Cartesian, x = r·cos(θ) and y = r·sin(θ); you can supply the angle in degrees or radians and the converter handles the unit internally.
Worked example
Convert the Cartesian point (3, 4) to polar coordinates.
- Radius: r = √(3² + 4²) = √(9 + 16) = √25 = 5.
- Angle: θ = atan2(4, 3) ≈ 0.9273 radians.
- In degrees: 0.9273 × 180 / π ≈ 53.13°.
The polar form is r = 5 at an angle of about 53.13° (0.9273 radians).
Frequently asked questions
- Why use atan2 instead of arctan(y/x) for the angle?
- Plain arctan cannot tell which quadrant the point is in and fails when x is zero. The atan2 function uses the signs of both x and y to return the correct angle across the full range from −180° to 180°.
- What angle does the origin (0, 0) return?
- At the origin the radius is zero and the angle is reported as 0 by convention, since a point with no distance from the origin has no meaningful direction.
- Can I work in radians instead of degrees?
- Yes. When converting polar to Cartesian you can choose to enter the angle in radians, and the Cartesian to polar direction reports the angle in both degrees and radians.
- Why must the radius be non-negative?
- This converter uses the standard convention that r is a distance, so it must be zero or positive. Direction is carried entirely by the angle θ rather than by a negative radius.