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Cartesian to Polar Converter

Enter the x and y coordinates to convert to polar form.

How it is calculated

For a point (x, y) the radius is r = √(x² + y²), the straight distance from the origin. The angle comes from θ = atan2(y, x), which uses the signs of both x and y to find the right quadrant. The tool then moves any negative angle into the range 0 to 360° (0 to 2π rad) by adding a full turn, so you always get a positive angle measured anticlockwise from the positive x-axis, rather than the −180° to 180° range many calculators and programming languages return. Radius is shown to four decimals, the angle in degrees and in radians. The origin (0, 0) gives r = 0 and θ = 0, although the angle there is really undefined.

Worked example

Convert the point (−3, −4). The radius is √((−3)² + (−4)²) = √(9 + 16) = √25 = 5.0000. The reference angle is arctan(4 ÷ 3) = 53.1301°. Because both coordinates are negative the point lies in the third quadrant, so the angle is 180° + 53.1301° = 233.1301° (atan2 itself gives −126.8699°, and adding 360° gives the same value). In radians that is 4.0689 rad. The tool shows r = 5.0000, θ = 233.1301° and 4.0689 rad.

Try this example →

Common mistakes

  • Using a plain arctangent of y ÷ x. For (−3, −4) that gives 53.13°, which is the first-quadrant angle; the point is actually at 233.13°. The quadrant has to come from the signs of x and y.
  • Expecting a negative angle for points below the x-axis. This tool reports 0 to 360°, so a point at −45° is shown as 315°; subtract 360 if you need the signed form.
  • Mixing degrees and radians. The calculator gives both, but a formula or calculator you copy the angle into must be set to the same unit.
  • Swapping x and y. The first value is the horizontal coordinate; entering the order as (y, x) mirrors the point and gives the angle 90° minus the correct one.
  • Reading the angle at the origin as meaningful. For (0, 0) the radius is 0 and the angle is shown as 0, but any angle is equally valid there.

About this calculator

This converter takes a point given in Cartesian coordinates (x, y) and expresses it in polar form (r, θ). The radius is r = √(x² + y²) — the straight-line distance from the origin — and the angle is θ = atan2(y, x), which correctly places the angle in the right quadrant unlike a plain arctangent. Results are shown in both degrees and radians since engineering and physics contexts commonly use one or the other.

Polar form is the natural language of rotation and periodic motion, so electrical engineers use it to express AC impedance and phasors, robotics and navigation engineers use it for range-and-bearing sensor data, and students and programmers rely on it when working with complex numbers, trigonometric graphing, or 2D game and graphics math where positions are more naturally described by distance and direction than by grid coordinates.

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Comments (1)

  • Editorial team

    The angle comes out between 0 and 360 degrees, not the minus 180 to 180 that atan2 gives in most programming languages, so subtract 360 from any angle above 180 to compare. The usual trap is plain arctan of y over x: for (-3, -4) it suggests 53.13 degrees, but the point sits at 233.13. Which quadrant is your point in?

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