Spherical Coordinate Converter

Cartesian vs. spherical coordinates

A point in three-dimensional space can be written in several coordinate systems. In Cartesian coordinates, a point is specified by its signed distances along three perpendicular axes: (x, y, z).

In spherical coordinates, the same point is described using a distance from the origin and two angles:

• Radial distance r ≥ 0: how far the point is from the origin.
• Polar angle θ: the angle between the positive z-axis and the line from the origin to the point.
• Azimuthal angle φ: the angle in the xy-plane from the positive x-axis toward the positive y-axis.

This convention is standard in many physics and engineering applications, especially where there is rotational or spherical symmetry (for example, around a point charge, a planet, or a spherical antenna pattern).

Conversion formulas

The converter uses the following standard relationships between Cartesian and spherical coordinates.

From Cartesian (x, y, z) to spherical (r, θ, φ)

Given a point (x, y, z):

• r = √(x² + y² + z²)
• θ = arccos(z / r) (for r > 0)
• φ = atan2(y, x)

In MathML form, the radial distance is:

From spherical (r, θ, φ) to Cartesian (x, y, z)

Given spherical coordinates (r, θ, φ) with r ≥ 0:

• x = r · sin(θ) · cos(φ)
• y = r · sin(θ) · sin(φ)
• z = r · cos(θ)

Typical angle ranges and conventions

There are several possible spherical coordinate conventions. This converter follows the widely used physics convention:

• Radius: r ≥ 0.
• Polar angle: 0 ≤ θ ≤ π (from the positive z-axis down to the negative z-axis).
• Azimuthal angle: usually considered in the range 0 ≤ φ < 2π in theory, but the underlying atan2(y, x) function used by the calculator may return values in (−π, π]. These values are equivalent modulo 2π.

If you supply angles outside these ranges, the point is still valid because the trigonometric functions are periodic. For example, φ = 3π and φ = π represent the same azimuthal direction.

All inputs and outputs for angles in this tool are in radians. To convert degrees to radians, multiply by π / 180. To convert radians to degrees, multiply by 180 / π.

How to use the converter

Convert from Cartesian to spherical

1. Enter your x, y, and z values in the Cartesian input fields.
2. Leave the spherical fields blank or ignore existing values.
3. Choose the option to convert to spherical coordinates.
4. Read the computed r, θ, and φ values (in radians).

Convert from spherical to Cartesian

1. Enter your r, θ, and φ values in the spherical input fields. Use radians for angles.
2. Leave the Cartesian fields blank or ignore existing values.
3. Choose the option to convert to Cartesian coordinates.
4. Read the computed x, y, and z values.

Worked examples

Example 1: Cartesian to spherical

Consider the point (x, y, z) = (1, 1, √2).

1. Compute the radius: r = √(1² + 1² + (√2)²) = √(1 + 1 + 2) = √4 = 2.
2. Compute the polar angle: θ = arccos(z / r) = arccos(√2 / 2). Since cos(π/4) = √2 / 2, we get θ = π / 4.
3. Compute the azimuthal angle with atan2: φ = atan2(1, 1) = π / 4, because both x and y are positive and equal.

Thus, the spherical coordinates are (r, θ, φ) = (2, π/4, π/4).

Example 2: Spherical to Cartesian

Now start with spherical coordinates (r, θ, φ) = (2, π/3, π/4).

1. Use x = r · sin(θ) · cos(φ): x = 2 · sin(π/3) · cos(π/4) = 2 · (√3/2) · (√2/2) = (2 · √3 · √2) / 4 = √6 / 2.
2. Use y = r · sin(θ) · sin(φ): y = 2 · sin(π/3) · sin(π/4) = 2 · (√3/2) · (√2/2) = √6 / 2.
3. Use z = r · cos(θ): z = 2 · cos(π/3) = 2 · (1/2) = 1.

The corresponding Cartesian point is (x, y, z) = (√6/2, √6/2, 1).

Interpreting your results

• If r is large, the point lies far from the origin; if r is small, the point is near the origin.
• When θ ≈ 0, the point is close to the positive z-axis; when θ ≈ π, it lies near the negative z-axis.
• When φ is between 0 and π/2, the point is in the first quadrant of the xy-plane (both x and y positive). Other intervals of φ reflect the different quadrants.

Note that multiple spherical representations can describe the same Cartesian point because adding integer multiples of 2π to φ or reflecting θ in specific ways can produce equivalent directions. The converter reports one consistent choice based on standard inverse trigonometric functions.

Comparison: Cartesian and spherical coordinates

Who this tool is for

• Students checking homework or gaining intuition for 3D coordinates.
• Engineers working with antenna patterns, acoustic fields, or 3D simulations.
• Scientists dealing with gravitational, electric, or quantum-mechanical problems.
• Developers implementing 3D graphics, ray tracing, or geometry utilities.

Assumptions and limitations

• Angles in radians only: All angle inputs and outputs are in radians. If you have degrees, convert them before entering.
• Origin behavior: At r = 0 (the origin), the angles θ and φ are mathematically undefined. The converter may return specific default values or treat them as zero, but any angle pair represents the same physical point.
• Radius sign: The standard convention uses r ≥ 0. Negative radii are not physically meaningful in this system and should not be used as inputs.
• Numerical precision: For very large or very small coordinate values, rounding and floating‑point precision can introduce small numerical errors in the results.
• Angle wrapping: The azimuth φ may be reported in the range (−π, π]. If you prefer [0, 2π), you can add 2π to negative values without changing the represented direction.
• Convention choice: This tool uses the physics convention (θ from the z-axis, φ in the xy-plane). Some mathematics texts swap the roles of θ and φ. Be sure to match the convention expected in your course or field.

Frequently asked questions

Are the angles in radians or degrees?

All angles in this spherical coordinate converter are in radians. If your values are in degrees, convert them first using radians = degrees × π / 180.

Which convention for θ and φ does this tool use?

The tool uses the common physics convention: θ is the polar angle measured from the positive z-axis (0 to π), and φ is the azimuthal angle in the xy-plane measured from the positive x-axis.

Can I enter negative angles?

Yes. Negative angles are allowed and are interpreted using periodic trigonometric functions. For example, φ = −π/4 is equivalent to φ = 7π/4.

What happens at the origin (r = 0)?

At the origin, the point is located exactly at (0, 0, 0). The angles θ and φ do not affect the position and are mathematically undefined. Any angle values represent the same point.

How accurate are the results?

The results are computed using standard floating‑point arithmetic. For typical educational and engineering uses, the accuracy is more than sufficient. Very extreme values may show minor rounding differences.