Spherical Coordinates Calculator - Rho, Theta, and Phi
Use this spherical coordinates calculator to switch between Cartesian (x, y, z) and spherical (rho, theta, phi) with degree or radian angle modes and live per-axis results.
Spherical Coordinates Calculator
Results
What Is a Spherical Coordinates Calculator?
A spherical coordinates calculator is a 3D geometry tool that converts a point between Cartesian (x, y, z) and spherical (rho, theta, phi) coordinates in either direction. The radius rho is the distance from the origin, the azimuth theta sits in the xy-plane from the positive x-axis, and phi is the polar angle from the positive z-axis. Common jobs include solving triple integrals, plotting antenna patterns, and reading a polar angle from a 3D scanner.
- • Triple integrals: Switch to spherical coordinates when an integrand depends on the distance from the origin, like electric field magnitudes in central potentials.
- • Antenna and beam patterns: Read beamwidth and pointing direction as theta and phi, then back out the Cartesian unit vector for steering the antenna.
- • Robotics and 3D scanning: Convert scanner outputs in (rho, theta, phi) into the (x, y, z) cloud a CAD or meshing tool expects.
- • Reference frames: Translate between geodetic latitude, longitude, altitude, and the local east-north-up frame in a spherical approximation.
The default convention is the math one: theta is the azimuth in the xy-plane and phi is the polar angle from the +z axis. Some physics texts swap the two names; this calculator keeps the math convention to match calculus textbooks and the standard Jacobian rho^2 sin(phi).
For the 2D version, the Cartesian to Polar Calculator covers the radius and a single angle in one row.
How the Spherical Coordinates Calculator Works
- rho: Radial distance from the origin, in the same length unit as x, y, z.
- theta: Azimuth angle in the xy-plane, measured counterclockwise from the positive x-axis.
- phi: Polar angle from the positive z-axis. 0 is the +z axis, pi (or 180 deg) is the -z axis.
- x, y, z: Cartesian coordinates of the same point. z is the height, x and y are the horizontal coordinates.
- radial projection: Helper value rho sin(phi), equal to sqrt(x^2 + y^2).
The Cartesian to spherical branch uses the Pythagorean theorem for rho, the two-argument arctangent atan2(y, x) for theta, and the inverse cosine of z over rho for phi. atan2 handles every quadrant correctly, which is why the single-argument arctan cannot replace it.
Example 1: Cartesian (3, 4, 12) to spherical
x = 3, y = 4, z = 12, unit = degrees
rho = sqrt(9 + 16 + 144) = 13. theta = atan2(4, 3) = 53.1301 deg. phi = acos(12 / 13) = 22.6199 deg.
rho = 13, theta = 53.1301 deg, phi = 22.6199 deg
The point sits 53 degrees around from the +x axis and tilts only 22.6 degrees off the +z axis, matching z = 12 being the largest coordinate.
Example 2: Spherical (5, 30 deg, 60 deg) to Cartesian
rho = 5, theta = 30 deg, phi = 60 deg
x = 5 sin(60) cos(30) = 3.75. y = 5 sin(60) sin(30) = 2.1651. z = 5 cos(60) = 2.5.
x = 3.75, y = 2.1651, z = 2.5
All three Cartesian components are positive, placing the point in the first octant.
According to Wolfram MathWorld, spherical coordinates describe a point in 3D by its radial distance rho, azimuth angle theta in the xy-plane, and polar angle phi from the positive z-axis, with the conversion formulas rho = sqrt(x^2 + y^2 + z^2), theta = atan2(y, x), and phi = acos(z / rho).
When the spherical coordinates describe a sphere (a constant rho), the Sphere Equation Calculator is the natural next step and accepts the standard and general forms of the sphere equation.
Key Concepts Explained
Four ideas show up in every spherical coordinate problem. Understanding each one makes the output much easier to interpret.
Rho as the distance from the origin
Rho is the Euclidean length of the position vector, equal to sqrt(x^2 + y^2 + z^2). It is always non-negative, and it is the only spherical quantity with units of length.
Theta as the azimuth in the xy-plane
Theta is the planar angle measured counterclockwise from the positive x-axis to the projection of the point on the xy-plane, wrapped into the full 0 to 360 degree (or 0 to 2 pi radian) circle.
Phi as the polar angle from the +z axis
Phi is the angle from the positive z-axis down to the position vector. Zero phi is the +z axis, 90 degrees is the xy-plane, 180 degrees is the -z axis.
Why the math convention uses theta and phi in that order
Calculus textbooks and ISO 80000-2 both use rho first, then theta (azimuth), then phi (polar). The calculator follows the math convention so the Jacobian rho^2 sin(phi) matches.
Once a point is in Cartesian form, the 3D Distance Calculator applies the Euclidean 3D distance formula.
How to Use This Spherical Coordinates Calculator
Pick the side you already have, choose the angle unit, and read the converted coordinates in the result panel.
- 1 Pick the source side: Open the Compute From dropdown and choose 'From Cartesian' for x, y, z or 'From Spherical' for rho, theta, phi. The other side becomes the output.
- 2 Choose degrees or radians: Set the Angle Unit dropdown to match how theta and phi are written. The unit applies to both the spherical inputs and outputs; the Cartesian x, y, z stay numeric.
- 3 Enter the three source values: Type the source values into the three matching fields. Negative rho is rejected, and phi is clamped to [0, 180] degrees (or [0, pi] radians). The matching min and max on the inputs track the unit dropdown.
- 4 Read the converted coordinates: The result panel shows the six numeric outputs plus the radial projection rho sin(phi). The chosen angle unit appears next to theta and phi.
- 5 Switch sides to verify: Flip the Compute From dropdown to the other side. The same point should reproduce with the same rho and x, y, z to within rounding.
- 6 Use the radial projection as a sanity check: The radial projection equals sqrt(x^2 + y^2). If the two numbers disagree, the angle unit or one of the inputs is wrong.
A sensor reports a target at rho = 13 m, theta = 53.13 deg, phi = 22.62 deg. Switch the source to 'From Spherical' and read the Cartesian (3, 4, 12).
Once a point is in Cartesian form, the Vector Magnitude Calculator confirms that the 3D magnitude equals rho, which is a quick round-trip check on the conversion.
Benefits of Using This Spherical Coordinates Calculator
A dedicated spherical tool removes the trigonometry overhead and prevents the atan vs atan2 bug. The benefits below explain why a calculator beats a notebook.
- • Bidirectional conversion in one tool: Cartesian to spherical and spherical to Cartesian share the same form, so there is no need to flip between two calculators.
- • Degree and radian support: A single toggle changes both theta and phi at once, so physics problems in radians and engineering problems in degrees use the same code path.
- • atan2 built in: The azimuth uses the two-argument arctangent, which correctly handles the third and fourth quadrants where the single-argument arctan fails.
- • Radial projection helper value: The result panel reports rho sin(phi) alongside the angles, which is the radius of the circle the point traces in the xy-plane.
- • Live recalculation on every keystroke: The output refreshes as soon as any input changes, so there is no need to click a separate 'convert' button.
- • Clamps out-of-range phi automatically: Phi outside [0, 180] degrees (or [0, pi] radians) is clamped to the valid range, and theta is wrapped into the standard interval so common typos do not blow up the result.
The Coordinates Converter steps through several 2D coordinate pairs at once.
Factors That Affect Your Results
Spherical coordinate formulas are exact, but the chosen angle unit, the quadrant, and the value of phi at the poles change the final numbers in predictable ways. These factors and limitations are the things to double-check before trusting a result.
Angle unit choice
Entering theta = 30 with the unit set to radians inflates the azimuth by a factor of 57.3 and produces a Cartesian point that is roughly 50 times too large in x and y. Always confirm the unit dropdown.
Quadrant of the point
A point in the third quadrant (x and y negative) has a positive theta in [180, 360) degrees. The single-argument arctan would give a negative angle, so the calculator uses atan2 and then wraps.
Phi at the poles
When phi is exactly 0 or 180 degrees, the point lies on the +z or -z axis. The radial projection collapses to 0, and theta is set to 0 by convention because the azimuth circle shrinks to a point.
Origin singularity
When x, y, z are all 0, the calculator returns rho = 0 and both theta and phi default to 0. Phi is undefined at the origin because z / rho is 0 / 0, so the displayed 0 is a convention.
Out-of-range phi input
Typing 200 degrees for phi looks valid, but phi is only defined in [0, 180] degrees (or [0, pi] radians). The calculator clamps to the valid range, which can hide a sign or unit error.
- • The formulas assume the math convention with theta as the azimuth in the xy-plane and phi as the polar angle from the +z axis. Physics texts sometimes swap the two angle names; users following the physics convention should rename the inputs.
- • The tool treats phi = 0 as the +z axis and phi = 180 as the -z axis. It does not support the colatitude convention used in some astronomy and geodesy references.
- • The result is exact to within the displayed four-decimal precision. Round-trip conversion can drift by up to 1e-4 in the least-significant digits because of floating-point rounding.
According to Wikipedia, spherical coordinates use the radius rho, the polar angle phi from the positive z-axis, and the azimuth angle theta from the positive x-axis in the xy-plane, with phi in [0, pi] and theta in [0, 2 pi).
The Sphere Calculator is a useful complement for problems where the spherical coordinates describe a sphere (a constant rho), and the Sphere Equation Calculator generalizes that to spheres offset from the origin.
Frequently Asked Questions
Q: What are spherical coordinates and how are they written?
A: Spherical coordinates describe a point in 3D using three numbers: rho is the radial distance from the origin, theta is the azimuth angle in the xy-plane from the positive x-axis, and phi is the polar angle from the positive z-axis. The triple (rho, theta, phi) names the same point as the Cartesian (x, y, z).
Q: What is the formula to convert Cartesian xyz to spherical rho theta phi?
A: Use rho = sqrt(x^2 + y^2 + z^2), theta = atan2(y, x) (wrapped to [0, 2 pi) when needed), and phi = acos(z / rho). The two-argument arctangent atan2 correctly handles points in every quadrant, and the inverse cosine places phi in [0, pi].
Q: How do you convert spherical to Cartesian coordinates?
A: Apply x = rho sin(phi) cos(theta), y = rho sin(phi) sin(theta), and z = rho cos(phi). The polar angle phi first projects rho onto the z-axis with cos(phi); the azimuth theta then rotates the remaining rho sin(phi) around the z-axis.
Q: What is the difference between the azimuth angle theta and the polar angle phi?
A: Theta is the planar angle in the xy-plane, measured counterclockwise from the positive x-axis; it behaves like the polar angle in 2D. Phi is the angle from the positive z-axis down to the position vector, so phi = 0 is the +z pole, phi = 90 degrees is the xy-plane, and phi = 180 is the -z pole.
Q: What are the valid ranges of rho, theta, and phi in spherical coordinates?
A: Rho is non-negative (rho >= 0). Theta is in [0, 360) degrees or [0, 2 pi) radians. Phi is in [0, 180] degrees or [0, pi] radians. The calculator clamps phi to that range and wraps theta into the standard interval.
Q: What is the equation of a sphere in spherical coordinates?
A: A sphere of radius R centered at the origin is simply rho = R. With phi restricted to [0, pi] and theta restricted to [0, 2 pi), the constant-rho surface covers the whole sphere, which is the main reason spherical coordinates simplify many integral problems.