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Spherical To Cartesian Coordinates Calculator

Work out spherical to cartesian coordinates instantly with clear inputs, formula shown and shareable results.

About Spherical To Cartesian Coordinates Calculator

Spherical coordinates use a radius, a polar angle θ measured down from the z-axis and an azimuth φ in the xy-plane. Converting gives x = r·sin θ·cos φ, y = r·sin θ·sin φ and z = r·cos θ.

Formula

Spherical to Cartesian

x = r·sin θ·cos φ, y = r·sin θ·sin φ, z = r·cos θ

Worked example

With the default values below, Spherical To Cartesian Coordinates Calculator returns x of 7.500000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Radius rminimum 0
10
Polar angle θ from the z-axisaccepts 0–180 °
60 °
Azimuthal angle φ in the xy-plane
30 °

Results

x
7.500000
y
4.330127
z
5.000000
Recovered radius
10.000000

Step by step

  1. 1Values used
  2. 2Spherical to Cartesian
  3. 3x= 7.500000
  4. 4y= 4.330127
  5. 5z= 5.000000
  6. 6Recovered radius= 10.000000

Assumptions and limits

  • The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For r = 10, θ = 60°, φ = 30°?

x = 10 × 0.8660254 × 0.8660254 = 7.5, y = 4.3301270 and z = 5.

Which angle is θ?

Conventions differ. Here θ is the polar angle from the z-axis, the physics convention; some maths texts swap θ and φ.

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