Convert between spherical (ρ, θ, φ), Cartesian (x, y, z), and cylindrical (r, θ, z) coordinates with step-by-step solutions, convention toggle, and projection diagrams.
Spherical coordinates describe a point in three-dimensional space using three values: the radial distance ρ from the origin, the azimuthal angle θ in the xy-plane from the positive x-axis, and the polar angle φ measured from the positive z-axis. This system is the natural choice whenever a problem has radial symmetry — gravitational fields, electromagnetic radiation, satellite orbits, and quantum mechanics wave functions all simplify dramatically in spherical form.
A common source of confusion is the two rival naming conventions used in physics and mathematics. Physicists typically use (ρ, θ, φ) where θ is azimuthal and φ is polar, while mathematicians swap the two angles. This calculator lets you toggle between both conventions so the labels always match your textbook.
The conversion formulas are compact. From spherical to Cartesian: x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ (physics convention). From Cartesian to spherical: ρ = √(x²+y²+z²), φ = arccos(z/ρ), θ = atan2(y, x). You can also go directly between spherical and cylindrical without touching Cartesian at all.
This tool handles all four conversion directions, displays every algebraic step, offers six presets for common test points (poles, equator, 45° cone), and renders interactive top-down and side projection diagrams so you can verify the point visually. A comparison table summarizes when each coordinate system is most appropriate.
Spherical-to-Cartesian conversion involves nested trig functions where swapping θ and φ (the physics vs. math convention issue) is the single most common source of error. This calculator handles both conventions, converts in all three directions (spherical ↔ Cartesian ↔ cylindrical), and shows seven output values so you can cross-check every component. Toggle between degrees and radians to match your textbook, and watch how the conversion formulas change when you switch from the physics convention (θ = azimuthal, φ = polar) to the math convention (θ = polar, φ = azimuthal).
Spherical → Cartesian (physics): x = ρ·sin(φ)·cos(θ) y = ρ·sin(φ)·sin(θ) z = ρ·cos(φ) Cartesian → Spherical: ρ = √(x²+y²+z²) φ = arccos(z/ρ) θ = atan2(y,x) Spherical ↔ Cylindrical: r_cyl = ρ·sin(φ) z = ρ·cos(φ) ρ = √(r²+z²) φ = atan2(r,z)
Result: x ≈ 1.732, y = 1, z ≈ 3.464
x = 4·sin(30°)·cos(60°) = 4·0.5·0.5 = 1. y = 4·sin(30°)·sin(60°) = 4·0.5·0.866 ≈ 1.732. z = 4·cos(30°) = 4·0.866 ≈ 3.464.
The most confusing aspect of spherical coordinates is the convention swap. In physics (ISO standard), θ is the azimuthal angle in the xy-plane and φ is the polar angle from the z-axis. In mathematics, these roles are reversed. Mixing conventions silently produces wrong answers because the formulas look identical. This calculator lets you toggle between the two, and the formula display updates accordingly, so you can see exactly which angle goes where.
The calculator supports three conversion directions: spherical → Cartesian, Cartesian → spherical, and spherical ↔ cylindrical. Direct formulas (x = ρ sinφ cosθ, etc.) are fastest, but the inverse direction requires care: φ = arccos(z/ρ) is straightforward, but θ = atan2(y, x) is needed to get the correct quadrant. The cylindrical shortcut r_cyl = ρ sinφ, z = ρ cosφ avoids the Cartesian intermediate entirely.
Spherical coordinates simplify any problem with radial symmetry. Gravitational and Coulomb fields depend only on ρ, making Laplace's equation separable. Antenna radiation patterns are expressed as gain G(θ, φ). In quantum mechanics, atomic orbitals are described by spherical harmonics Yₗᵐ(θ, φ), and the volume element ρ² sinφ dρ dφ dθ appears in every integral over a sphere.
In physics, θ is the azimuthal angle (xy-plane) and φ is the polar angle (from z-axis). In math, the roles of θ and φ are swapped.
Whenever the problem has spherical or radial symmetry — gravity, Coulomb law, radiation patterns, wave functions, and geodesy. Use this as a practical reminder before finalizing the result.
r_cyl = ρ·sin(φ), θ_cyl = θ (same azimuthal angle), z = ρ·cos(φ). No Cartesian step needed.
The point is at the origin; the angles are undefined (every direction from the origin looks the same). Keep this note short and outcome-focused for reuse.
It shows all three spherical values, three Cartesian values, and the cylindrical radial distance for easy cross-checking. Apply this check where your workflow is most sensitive.
Yes — a negative azimuthal angle simply counts clockwise instead of counter-clockwise. The calculator normalizes it to [0, 2π).