Instantly convert cylindrical coordinates (r, θ, z) to spherical coordinates (ρ, θ, φ). Use this free calculator for geometry, physics, engineering, robotics, computer graphics, and multivariable calculus.
Enter cylindrical coordinates and choose your preferred angle unit. This calculator uses the common spherical convention where φ is measured from the positive z-axis.
Cylindrical and spherical coordinates are alternative ways to represent points in three-dimensional space. In Cartesian form, a point is written as (x, y, z). In cylindrical form, the same point becomes (r, θ, z), where r is distance from the z-axis, θ is the azimuth around the z-axis, and z is the vertical coordinate. In spherical form, the point is represented as (ρ, θ, φ), where ρ is total distance from the origin, θ is usually the same azimuth angle, and φ is the angle measured down from the positive z-axis.
These systems are extremely useful when geometry has symmetry. Cylindrical coordinates are ideal for objects with rotational symmetry around an axis, such as pipes, shafts, cables, and many waveguide problems. Spherical coordinates are natural for radially symmetric domains such as planets, radiation fields, gravitational potentials, and 3D directional sensing.
Because both systems capture rotational structure, converting cylindrical to spherical coordinates is common in calculus, electromagnetics, fluid dynamics, and 3D simulation pipelines.
Given cylindrical coordinates (r, θ, z), the spherical coordinates under the common physics convention are:
This convention uses φ as the polar angle from +z, so φ is between 0 and π. Using atan2(r, z) instead of a simple arctangent handles sign and quadrant correctly, especially when z is negative.
Related transformations are:
| From Cylindrical | To Spherical |
|---|---|
| r | r = ρ sin φ |
| z | z = ρ cos φ |
| θ | θ unchanged |
If your textbook uses a math convention where θ and φ are swapped, the geometry is still correct. Only angle naming changes. Always check your course definition before submitting homework or integrating formulas into code.
To convert cylindrical to spherical coordinates quickly and accurately:
Because θ is unchanged, the major computational work is finding ρ and φ. Precision matters in numerical software, so engineering workflows often keep at least 6 decimal places internally and round only when displaying results.
Example A: Convert cylindrical point (r, θ, z) = (5, 30°, 12).
So the spherical coordinate is (13, 30°, 22.6199°).
Example B: Convert cylindrical point (3, 1.2 rad, -4).
So the spherical coordinate is (5, 1.2, 2.4981) in radians.
Coordinate conversion is not just a classroom exercise. Engineers and scientists switch coordinate systems to simplify equations, reduce computational cost, and align models with sensor data. Here are practical domains where cylindrical-to-spherical conversion appears regularly:
In many simulation workflows, one coordinate system is used for data collection and another for solving governing equations. A robust cylindrical to spherical calculator helps validate transformations before they are embedded in larger systems.
If you regularly switch conventions, create a short checklist for your team: angle naming, reference axis, angle range, and units. This eliminates most integration bugs in simulation and data-processing code.
Yes, in the common convention. Both coordinate systems use the same azimuth angle around the z-axis, so θ is carried over directly.
Because φ is measured from the positive z-axis toward the point. With r as the perpendicular component and z as the adjacent component, tan(φ) = r/z. Using atan2(r, z) preserves correct quadrants.
Standard cylindrical coordinates use r ≥ 0. Negative values can be represented by adjusting θ, but most engineering and math contexts keep r nonnegative.
The point lies on the z-axis. Then φ is 0 if z > 0, π if z < 0, and undefined at the origin where both r and z are zero.
Yes. It is useful for classroom work, quick verification, and pre-processing checks in technical projects.
Use the calculator at the top of this page anytime you need fast, accurate cylindrical to spherical conversion. Whether you are studying multivariable calculus or building a production model, consistent coordinate handling saves time and prevents costly downstream errors.