About the Spherical & Cylindrical Coordinates Converter
Three-dimensional points can be described in three standard ways: Cartesian coordinates (x, y, z), cylindrical coordinates (a horizontal distance from the z-axis, an azimuth angle around it, and a height z), and spherical coordinates (a distance from the origin plus two angles). This converter takes a point in any one of those systems and returns it in all three.
To avoid the notorious symbol clash between textbooks, inputs and outputs are labelled by meaning: the azimuthal angle is measured in the xy-plane counter-clockwise from the +x axis, and the polar angle is measured down from the +z axis (0° at the north pole, 90° on the equator, 180° at the south pole). Physics and ISO 80000-2 call these φ and θ respectively; many calculus books swap the letters.
It is handy for multivariable calculus and physics homework, setting up triple integrals, antenna and acoustics work, robotics and 3D graphics, and converting latitude/longitude-style directions into vectors. Angles can be entered and shown in degrees or radians.
How to use the spherical & cylindrical coordinates converter
- 1Choose the coordinate system your point is in now.
- 2Pick degrees or radians for the angles.
- 3Enter the three coordinates.
- 4Read the point in the other two systems, plus every individual value.
Formula and method
Cylindrical coordinates keep z and replace x and y with polar coordinates in the horizontal plane: ρ = √(x² + y²) and φ = atan2(y, x), with x = ρ cos φ and y = ρ sin φ going back. Spherical coordinates use the straight-line distance r = √(x² + y² + z²), the polar angle θ = arccos(z/r) measured from the positive z-axis, and the same azimuth φ.
The two-argument arctangent atan2(y, x) puts φ in the correct quadrant, and the result is normalised to 0–360° (0–2π). Because ρ = r sin θ and z = r cos θ, converting between cylindrical and spherical never needs x and y. On the z-axis the azimuth is undefined and is reported as 0. Elevation (latitude-style angle) is 90° − θ.
- x, y, z
- Cartesian coordinates
- ρ
- Cylindrical radius: distance from the z-axis
- r
- Spherical radius: distance from the origin
- θ
- Polar (zenith) angle from the +z axis, 0–180°
- φ
- Azimuth in the xy-plane from the +x axis, 0–360°
Worked examples
Cartesian (1, 1, 1) to spherical and cylindrical
r = √3 ≈ 1.7321 and ρ = √2 ≈ 1.4142. The azimuth is atan2(1, 1) = 45°, and the polar angle is arccos(1/√3) ≈ 54.74°, so the point sits about 35.26° above the xy-plane.
Spherical r = 2, θ = 60°, φ = 45° to Cartesian
x = 2 · sin 60° · cos 45° ≈ 2 × 0.8660 × 0.7071 ≈ 1.2247, y is the same, and z = 2 · cos 60° = 1. The cylindrical radius is r sin θ = √3 ≈ 1.7321.
Cylindrical ρ = 2, φ = 45°, z = 1 to spherical
r = √(ρ² + z²) = √5 ≈ 2.2361 and θ = arctan(ρ/z) = arctan 2 ≈ 63.43°. The azimuth stays 45°, and x = y = 2 cos 45° ≈ 1.4142.
Cartesian (−3, −4, 0) in radians
The point lies in the xy-plane in the third quadrant. r = 5, the polar angle is π/2 ≈ 1.5708 rad, and the azimuth is atan2(−4, −3) + 2π ≈ 4.0689 rad (about 233.13°).
Frequently asked questions
How do you convert Cartesian to spherical coordinates?+
Compute r = √(x² + y² + z²), the polar angle θ = arccos(z/r) and the azimuth φ = atan2(y, x). For (1, 1, 1): r ≈ 1.732, θ ≈ 54.74° and φ = 45°.
Why do some books swap θ and φ?+
Physics and the ISO 80000-2 standard use θ for the polar angle and φ for the azimuth, while many US calculus books use θ for the azimuth (to match polar coordinates) and φ for the polar angle. This tool labels angles by meaning to avoid the confusion.
How do you convert spherical to cylindrical coordinates?+
Keep the azimuth φ, then ρ = r sin θ and z = r cos θ. Going back, r = √(ρ² + z²) and θ = arctan(ρ/z) (or arccos(z/r)).
What is the difference between polar angle and elevation?+
The polar (zenith) angle is measured down from the +z axis, while elevation or latitude is measured up from the xy-plane. They always add to 90°, so a polar angle of 60° equals an elevation of 30°.
What is the azimuth of a point on the z-axis?+
It is undefined because x = y = 0, so every azimuth describes the same point. By convention this converter reports 0 in that case, and at the origin both angles are reported as 0.