Spherical to Cartesian Coordinates

Convert spherical coordinates (r, θ, φ) into Cartesian coordinates (x, y, z). Polar and azimuthal angles support both degrees and radians.

Input

Enter spherical coordinates (r, θ, φ) to convert them into Cartesian coordinates (x, y, z). θ is the polar angle from the z axis and φ is the azimuthal angle.

Radius r

Polar angle θ

Azimuthal angle φ

Result

Cartesian coordinates (x, y, z)

(3.061862, 3.061862, 2.5)

x component

3.061862

y component

3.061862

z component

2.5

Radius r

Polar angle θ

60° / 1.047198 rad

Azimuthal angle φ

45° / 0.785398 rad

Computed with x = r sinθ cosφ, y = r sinθ sinφ and z = r cosθ, where θ is the polar angle and φ is the azimuthal angle.

How it works

Converts spherical coordinates (r, θ, φ) into Cartesian coordinates (x, y, z). Here r is the radial distance from the origin, θ is the polar angle measured from the z axis (colatitude), and φ is the azimuthal angle measured in the xy plane from the x axis.
The conversion uses x = r sinθ cosφ, y = r sinθ sinφ, and z = r cosθ. The projection of the radius onto the xy plane is r sinθ, which φ then splits into the x and y components, while z comes directly from r cosθ.
Angles can be entered in degrees or radians. In degree mode the values are multiplied by π / 180 internally before the trigonometric functions are applied, and both degree and radian forms are shown in the result.
When θ is 0 the point lies on the z axis with z equal to r, and when θ is 90 degrees the point lies in the xy plane. A negative r produces the point in the opposite direction, following the formulas directly.
This physics convention, with θ as the polar angle and φ as the azimuthal angle, follows the ISO 80000 notation. Some mathematics texts swap the roles of θ and φ, so check what each input means in your source.

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