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.
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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.
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
5
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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