Calculator
Choose a conversion path, enter known values, and submit to view the converted coordinates, formulas, and 3D plot.
Example Data Table
| Case | Input System | Values | Converted Highlight |
|---|---|---|---|
| 1 | Cartesian | x = 3, y = 4, z = 5 | ρ = 7.0711, θ = 53.1301°, φ = 45° |
| 2 | Spherical | ρ = 10, θ = 30°, φ = 60° | x = 7.5, y = 4.3301, z = 5 |
| 3 | Cylindrical | r = 6, θ = 45°, z = 8 | ρ = 10, φ = 36.8699° |
| 4 | Spherical | ρ = 12, θ = 120°, φ = 75° | r = 11.5911, z = 3.1058 |
Formula Used
Cartesian to Spherical
ρ = √(x² + y² + z²) θ = atan2(y, x) φ = arccos(z / ρ)Spherical to Cartesian
x = ρ sin(φ) cos(θ) y = ρ sin(φ) sin(θ) z = ρ cos(φ)Cylindrical to Spherical
ρ = √(r² + z²) θ = θ φ = arccos(z / ρ)Spherical to Cylindrical
r = ρ sin(φ) θ = θ z = ρ cos(φ)These formulas assume the physics-style spherical system, where φ is measured from the positive z-axis and θ is measured in the x-y plane.
How to Use This Calculator
- Select the conversion mode you need.
- Choose degrees or radians for angle entry and output.
- Enter the known coordinate values in the visible input fields.
- Set your preferred decimal precision.
- Click Convert Coordinates to generate the result.
- Review the summary tables, step-by-step calculations, and 3D graph.
- Download the result as CSV or PDF when needed.
FAQs
1. What is spherical coordinate conversion?
It changes a point’s representation between spherical, Cartesian, and cylindrical systems without changing the actual geometric location in three-dimensional space.
2. What does ρ represent here?
ρ is the radial distance from the origin to the point. It is always non-negative in this calculator.
3. What is the difference between θ and φ?
θ is the azimuth angle in the x-y plane. φ is the polar angle measured downward from the positive z-axis.
4. Can I use radians instead of degrees?
Yes. Choose radians in the angle unit selector. Inputs and displayed angular results will then use radians.
5. Why does the tool use atan2 for θ?
atan2 correctly determines the angle’s quadrant from x and y signs. That avoids ambiguity that appears with a basic arctangent.
6. Why is my radius not allowed to be negative?
A non-negative radius keeps the representation standard and easier to interpret. Negative radii can produce equivalent but less intuitive descriptions.
7. What does the 3D graph show?
The graph shows the final point in Cartesian space. It helps verify location, direction, and sign consistency after conversion.
8. Is this useful for vector calculus and geometry?
Yes. It is useful for surfaces, volumes, multivariable calculus, physics, engineering, and coordinate transformations involving symmetry around an origin.