Introduction to spherical-coordinate conversion
This spherical coordinate converter lets you describe the same 3D point in two different languages: the component-based Cartesian form and the radius-and-angle spherical form. Cartesian coordinates list how far a point sits along the x, y, and z axes, while spherical coordinates replace those three offsets with one distance from the origin and two directions. That shift is especially helpful when the geometry is centered on a point, a shell, or a radial field.
The calculator follows the physics and engineering convention. Here, r is the distance from the origin, θ (theta) is the polar angle measured down from the positive z-axis, and φ (phi) is the azimuth measured in the xy-plane from the positive x-axis. That is the convention used in the formulas on this page and in the result fields below. It is worth checking because some books reverse the names of θ and φ, and a mismatch in notation can make two correct answers look different.
Use the converter when you want to switch quickly between the two coordinate systems without doing the trigonometry by hand. Enter Cartesian values to get a spherical description, or enter spherical values to recover the Cartesian components. The page also shows the formulas, a concrete example, and a few notes about rounding, the origin, and angle conventions so you can interpret the output with confidence. All angles are shown in radians.
How to use the spherical coordinate converter
To convert a point from Cartesian to spherical coordinates, type x, y, and z into the first set of fields and press Convert to Spherical. The calculator will compute the radius, the polar angle, and the azimuth, then fill the spherical fields with decimal values. This is the right direction when a problem starts with coordinates on three perpendicular axes but you need a radial description of the same point.
To convert a point from spherical to Cartesian coordinates, enter r, θ, and φ in the second set of inputs and press Convert to Cartesian. The calculator will project the radius onto the xy-plane, split that projection into x and y, and recover the vertical component as z. This direction is useful when you are graphing, checking a simulation, or translating a textbook answer into rectangular components.
The outputs are rounded to four decimal places, which is usually enough for classwork and quick verification. If you are comparing against an exact expression such as π/4 or √6/2, remember that this calculator returns decimal approximations rather than symbolic forms. After a successful conversion, you can also copy the displayed result text with the Copy Result button.
Formula for spherical-coordinate conversion
The spherical coordinate converter begins by measuring the straight-line distance from the origin to the point. That distance becomes r, and the remaining angles are derived from the direction of the point relative to the positive z-axis and the positive x-axis in the xy-plane. The calculator uses atan2(y, x) for φ so that the correct quadrant is preserved.
For a point (x, y, z), the radius is given by the relation below.
Once r is known, θ is computed as arccos(z / r) when r > 0, and φ is computed with atan2(y, x). In the other direction, the calculator reconstructs Cartesian coordinates by taking the radius, leaning it away from the z-axis by θ, and then rotating that horizontal projection by φ. The formulas x = r · sin(θ) · cos(φ), y = r · sin(θ) · sin(φ), and z = r · cos(θ) are the standard conversion rules used here.
Those equations match the geometry of a sphere centered at the origin. r controls size, θ controls tilt away from the vertical axis, and φ controls orientation around the vertical axis. Together they locate the same point that Cartesian coordinates describe as x, y, and z offsets.
Coordinate meaning and angle conventions for spherical coordinates
Spherical coordinates are easiest to picture if you separate distance from direction. Imagine first choosing a sphere centered at the origin: r picks the sphere’s radius. Then imagine a ray from the origin to the point: θ tells you how far that ray drops from the positive z-axis. Finally, φ tells you how that ray turns around the z-axis after you project it into the xy-plane.
In this calculator, the usual ranges are r ≥ 0 and 0 ≤ θ ≤ π, while φ is naturally wrapped by atan2(y, x). That means the displayed azimuth may appear in (−π, π], and that is still the same direction you would get from an angle shifted by 2π. For example, −π/4 and 7π/4 point toward the same horizontal direction even though they look different on paper.
Because the calculator is radian-based, any degree input must be converted before you type it in. Multiply degrees by π / 180 to obtain radians, or multiply radians by 180 / π to report them in degrees. If you skip that step, the formulas will still run, but the point you get back will not match the geometry you intended.
Worked example: converting a Cartesian point to spherical form
A simple spherical-coordinate example starts with the Cartesian point (1, 1, √2). The radius comes from the distance formula: r = √(1² + 1² + (√2)²) = √4 = 2. With that radius in hand, the polar angle is θ = arccos(z / r) = arccos(√2 / 2), which equals π/4.
The azimuth is φ = atan2(1, 1) = π/4, so the spherical form is (2, π/4, π/4). This is a good example of how a spherical description can make symmetry easy to see: the point sits on a radius of length 2, halfway between the positive x- and y-directions, and halfway down from the positive z-axis.
You can reverse the same example as a consistency check. Starting with (r, θ, φ) = (2, π/3, π/4), the formulas give x = 2 · sin(π/3) · cos(π/4) = √6 / 2, y = 2 · sin(π/3) · sin(π/4) = √6 / 2, and z = 2 · cos(π/3) = 1. The recovered Cartesian point is therefore (√6/2, √6/2, 1), which matches the sort of decimal result the calculator will display.
Interpreting the spherical-coordinate result
After a spherical-coordinate conversion, each value describes a different part of the point’s geometry. A larger r means the point lies farther from the origin, while a smaller r means it is closer to the center. If θ is near 0, the point lies close to the positive z-axis; if θ is near π, it lies close to the negative z-axis; and if θ is near π/2, it sits near the xy-plane.
The azimuth φ tells you how the point is rotated around the vertical axis. Values near 0 point toward positive x, values near π/2 point toward positive y, and the other quadrants are handled automatically by atan2. That is why the calculator can distinguish between directions that would otherwise look similar if you only used a plain arctangent.
More than one spherical description can represent the same point. Adding 2π to φ does not change the location, and there are other equivalent ways to write angles that differ by full turns. The calculator reports one consistent numeric answer so you have a single value to compare, copy, or enter into a later calculation.
Applications and comparison with Cartesian coordinates in spherical conversion
Spherical coordinates are most useful when a problem is organized around a center rather than a grid. They show up naturally in gravity, electrostatics, radiation patterns, wave motion, astronomy, and any setting where distance from a source matters more than separate x, y, and z offsets.
Cartesian coordinates are still the better choice when motion follows straight axes or when a program expects rectangular components. Spherical coordinates are better when symmetry is radial, because a sphere is written simply as r = constant instead of x² + y² + z² = r². In calculus, that same symmetry also explains why the volume element changes to r² sin(θ) · dr · dθ · dφ.
Because of that, this converter is useful both as a homework aid and as a conceptual bridge. If you are reading a formula sheet, building a simulation, or trying to decide which coordinate system makes a geometry easiest to handle, a quick conversion can reveal whether the structure of the problem is really radial, angular, or component-based.
Limitations and assumptions of the spherical coordinate converter
This spherical coordinate converter assumes the standard convention used throughout the page: r is nonnegative, θ is measured from the positive z-axis, and φ is measured in the xy-plane from the positive x-axis. If your textbook or software swaps the names of the angles, the underlying point may still be the same even though the labels do not match.
The origin is the main edge case. When r = 0, the point is exactly at (0, 0, 0), and the angles are not uniquely defined because every direction points to the same place. In that situation, a numerical converter cannot assign a meaningful direction, so the angles should be treated as undefined rather than interpreted too literally.
The converter also returns decimal approximations instead of symbolic expressions. That makes it practical for fast checking, but it means a result such as 0.7854 should be read as an approximation to π/4 rather than as an exact value. Ordinary floating-point rounding can also appear when inputs are very large, very small, or close to a coordinate axis.
If you need exact symbolic simplification or a different angle convention, you will need to transform the result by hand or with a CAS. For everyday coordinate conversion, though, the numerical output is usually enough to confirm the point and its direction.
Ray Aim: a spherical-coordinates arcade game
Practice thinking in angles instead of components. A glowing target lands somewhere on the sphere at its own polar angle θ and azimuth φ. Swing your ray to the same direction and fire. Use the arrow keys for fine control — Up/Down tilts θ away from the z-axis, Left/Right spins φ around it — then press Space to fire. On a touch screen, just tap the sphere where you want to aim. The closer your ray lines up with the target direction, the more points the hit is worth.
Click to play. Match the target's θ and φ, then fire before the clock runs out.
Why this trains the right intuition: the target and your ray are the same length, so nothing about the puzzle depends on r. What separates a miss from a bullseye is purely direction — exactly the two numbers, θ and φ, that spherical coordinates isolate. After a few rounds you start reading a point's tilt and spin at a glance, which is the same skill that makes the converter above feel obvious rather than mechanical.