Spherical Coordinates Calculator

Convert between Cartesian (rectangular) and spherical coordinates in 3D space. Enter your X, Y, Z values to get radial distance (ρ), polar angle (φ), and azimuthal angle (θ) — or go the other way by entering spherical coordinates to recover Cartesian form. Angles are shown in both degrees and radians.

X-coordinate of the point

Y-coordinate of the point

Z-coordinate of the point

Distance from origin to point

°

Angle from positive X-axis in XY plane (0° to 360°)

°

Angle from positive Z-axis (0° to 180°)

Results

ρ (Radial Distance)

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θ — Azimuthal Angle (degrees)

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θ — Azimuthal Angle (radians)

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φ — Polar Angle (degrees)

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φ — Polar Angle (radians)

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X (Cartesian)

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Y (Cartesian)

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Z (Cartesian)

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Frequently Asked Questions

What are spherical coordinates?

Spherical coordinates describe a point in 3D space using three values: ρ (radial distance from the origin), θ (azimuthal angle in the XY plane measured from the positive X-axis), and φ (polar angle measured from the positive Z-axis). They are especially useful in physics and engineering problems involving spherical symmetry.

How do you convert Cartesian coordinates to spherical coordinates?

Given a point (X, Y, Z), the spherical coordinates are computed as: ρ = √(X² + Y² + Z²), θ = atan2(Y, X) (adjusted to range 0–360°), and φ = arccos(Z / ρ). The azimuthal angle θ ranges from 0 to 2π (0° to 360°) and the polar angle φ ranges from 0 to π (0° to 180°).

How do you convert spherical coordinates back to Cartesian?

Given (ρ, θ, φ), the Cartesian coordinates are: X = ρ · sin(φ) · cos(θ), Y = ρ · sin(φ) · sin(θ), and Z = ρ · cos(φ). Make sure your angles are in radians when applying these formulas programmatically.

What is the difference between the azimuthal angle θ and the polar angle φ?

The azimuthal angle θ (theta) is measured in the horizontal XY plane from the positive X-axis, ranging from 0° to 360°. The polar angle φ (phi) is measured from the positive Z-axis downward toward the XY plane, ranging from 0° to 180°. Together they define the direction of the point from the origin.

Can this calculator handle 2D coordinates (no Z component)?

Yes. Simply enter Z = 0 for a 2D point in the XY plane. The radial distance ρ will equal the 2D distance from the origin, the azimuthal angle θ will give the standard polar angle, and the polar angle φ will be 90° since the point lies in the equatorial plane.

Why does the calculator return θ = 0° when both X and Y are zero?

When X = 0 and Y = 0, the point lies exactly on the Z-axis and the azimuthal angle θ is undefined mathematically. By convention, calculators typically return θ = 0° in this case, since any angle gives the same point on the Z-axis.

What units are the angles given in?

This calculator provides both degrees and radians for each angle (θ and φ). Degrees are the most intuitive for many users, while radians are standard in mathematical and programming contexts. You can use whichever unit suits your application.

What is the radial distance ρ and can it be negative?

The radial distance ρ (rho) is the straight-line distance from the origin (0, 0, 0) to the point, always equal to √(X² + Y² + Z²). It is always a non-negative value. A ρ of zero means the point is at the origin itself.

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