Free · Coordinate geometry calculators 📈

Cartesian & Polar Coordinates in Both Directions

Convert x/y coordinates to radius and angle, or radius and angle to x/y, with quadrant-aware angles and selectable angle units.

  • Formula & worked example
  • Private in your browser
  • No signup
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

Cartesian coordinates (0, 5) correspond to radius 5 and angle 90°. Polar coordinates (5, 180°) give Cartesian coordinates (−5, 0).

r = √(x² + y²); θ = atan2(y, x); x = r cos(θ); y = r sin(θ)

Worked example

Enter these known values and leave the other values blank.

Cartesian x input
0
Cartesian y input
5
Polar radius input
5
Polar angle input
180 deg
Computed radius
5
Computed principal angle
90 deg
Computed Cartesian x
-5
Computed Cartesian y
0

Assumptions and limitations

  • Use one common length unit for x, y and radius. The numeric coordinate fields do not perform length-unit conversion.
  • The two groups are independent forward conversions. Changing a calculated coordinate does not reverse-solve that group; use the other group for the opposite direction.
  • The Cartesian-to-polar angle is quadrant-aware, measured counterclockwise from the positive x-axis, and uses the atan2 principal range. Negative angles are valid.
  • At the origin, the angle has no unique mathematical value. This implementation displays zero as its atan2 convention; every angle describes radius zero.
  • Polar radius must be nonnegative. Polar angle input is bounded to ±1,000,000 radians for this finite-precision trigonometric calculation.

Common questions

Why not use arctan(y/x)?

atan2 uses the signs of both coordinates, handles the vertical axis and distinguishes opposite quadrants.

References

Bookify bounds polar angle entry to ±1,000,000 radians for finite-precision trigonometry.

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