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XYZ & Spherical Coordinates: Radius, Polar Angle & Azimuth

Convert between Cartesian xyz and spherical coordinates using a polar angle from +z and an azimuth around the xy-plane.

  • Formula & worked example
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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

The point (1, 1, √2) has radius 2, polar angle 45° from +z and azimuth 45° from +x. With radius 2, polar angle 90° and azimuth 0°, the reverse calculation gives (2, 0, 0).

r = √(x²+y²+z²); θ = acos(z/r); φ = atan2(y,x); x = r sin θ cos φ; y = r sin θ sin φ; z = r cos θ

Worked example

Enter these known values and leave the other values blank.

Spherical input radial distance
2
Spherical input polar angle from +z
90 deg
Spherical input azimuth from +x
0 deg
Calculated Cartesian x
2
Calculated Cartesian y
0
Calculated Cartesian z
0

Assumptions and limitations

  • This page uses θ for polar angle from +z and φ for azimuth from +x toward +y. Some references exchange those symbols or use elevation from the xy-plane instead; check the labels.
  • Radius is nonnegative, polar angle is between zero and 180°, and azimuth can be any equivalent angle. Cartesian x, y and z use one common length scale.
  • The two sections are independent forward conversions. Outputs do not automatically populate the other section.
  • At a nonzero point on the z-axis, polar angle is 0° or 180° and azimuth is undefined. At the origin both angles are undefined. Undefined output angles remain blank.
  • Small residuals around mathematically zero Cartesian components can reflect finite-precision trigonometric evaluation.

Common questions

Is the polar angle the same as elevation?

No. The polar angle is measured from +z. Elevation above the xy-plane equals 90° minus the polar angle.

Why is azimuth blank on the vertical axis?

Every azimuth represents the same point when x and y are zero. There is no unique horizontal direction.

References

Bookify leaves azimuth undefined on the vertical axis, including the origin. Bookify requires nonnegative spherical radius and a polar angle between zero and 180 degrees.

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