Introduction
This spherical coordinate converter helps you move between two common ways of describing a point in three-dimensional space. In Cartesian coordinates, a point is written as (x, y, z), which tells you how far the point lies along three perpendicular axes. In spherical coordinates, the same point is written as (r, θ, φ), which describes the point by its distance from the origin and two angles. The calculator is useful when a problem is easier to understand in terms of distance and direction rather than separate horizontal and vertical components.
The page follows the standard physics and engineering convention. Here, r is the radial distance from the origin, θ (theta) is the polar angle measured down from the positive z-axis, and φ (phi) is the azimuthal angle measured in the xy-plane from the positive x-axis. That convention matters because some textbooks swap the meanings of θ and φ. If you are checking homework, comparing software outputs, or reading a formula sheet, always confirm which convention is being used before you compare numbers.
This tool is designed to be practical as well as explanatory. You can enter Cartesian values and convert them to spherical form, or enter spherical values and convert them back to Cartesian form. The calculator also includes the core formulas, a worked example, and notes about assumptions and edge cases so the result is easier to trust and interpret. All angles in this calculator are in radians, not degrees.
How to use
Using the converter is straightforward. If you already know the point in Cartesian form, enter values for x, y, and z in the first group of inputs and click Convert to Spherical. The calculator will compute the radius r, the polar angle θ, and the azimuthal angle φ, then place those values into the spherical input fields. This is helpful when you want to understand a point in terms of distance from the origin and direction in space.
If you instead know the point in spherical form, enter r, θ, and φ in the second group of inputs and click Convert to Cartesian. The calculator will compute the corresponding x, y, and z values and place them into the Cartesian fields. This direction is useful when you need rectangular coordinates for graphing, simulation, geometry, or component-based calculations.
Because the calculator works numerically, it displays decimal approximations rounded to four decimal places. That is usually enough for classroom work, quick checks, and many engineering tasks. If you are comparing your result to an exact symbolic answer such as π/4 or √6/2, remember that the calculator will show the decimal form instead. Also note that the copy button lets you quickly copy the result text after a successful conversion.
Formula
The conversion from Cartesian coordinates to spherical coordinates starts by finding the straight-line distance from the origin to the point. That distance is the radius r. Once the radius is known, the polar angle θ is found from the ratio of the vertical coordinate to the radius, and the azimuthal angle φ is found from the horizontal direction in the xy-plane. The calculator uses the standard atan2(y, x) function for φ, which correctly identifies the quadrant.
For a point (x, y, z), the main relationship for the radius is shown below.
The remaining spherical values are computed with θ = arccos(z / r) for r > 0 and φ = atan2(y, x). In the other direction, when spherical coordinates are known, the calculator reconstructs the Cartesian components by projecting the radius onto the horizontal plane and then splitting that horizontal part into x and y. The formulas are x = r · sin(θ) · cos(φ), y = r · sin(θ) · sin(φ), and z = r · cos(θ).
These formulas are standard because they match the geometry of a sphere centered at the origin. The radius tells you how far away the point is. The angle θ tells you how far down from the positive z-axis the point lies. The angle φ tells you which direction the point faces when viewed from above in the xy-plane. Together, those three values identify the same point as (x, y, z).
Coordinate meaning and angle conventions
It helps to picture spherical coordinates in stages. First imagine a sphere centered at the origin. The value r chooses the size of that sphere. Next imagine a line from the origin to the point. The angle θ tells you how far that line tilts away from the positive z-axis. Finally, the angle φ tells you how the point is rotated around the z-axis when projected onto the xy-plane.
In this calculator, the usual ranges are r ≥ 0, 0 ≤ θ ≤ π, and an azimuth often interpreted over a full turn. However, because the calculator uses atan2(y, x), the returned value of φ may appear in the interval (−π, π]. That is not an error. A negative azimuth and a positive azimuth that differs by 2π describe the same direction. For example, −π/4 and 7π/4 point the same way in the plane.
All angle inputs and outputs are in radians. If your source values are in degrees, convert them before entering them. Multiply degrees by π / 180 to get radians, or multiply radians by 180 / π to get degrees. This matters because entering degree values directly into a radian-based calculator will produce the wrong point even if the formulas themselves are correct.
Example
Suppose you start with the Cartesian point (1, 1, √2). The radius is found from the distance formula: r = √(1² + 1² + (√2)²) = √4 = 2. Next, 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 example is a good reminder that spherical coordinates often reveal symmetry more clearly than Cartesian coordinates do.
Now reverse the process with spherical coordinates (r, θ, φ) = (2, π/3, π/4). Using the standard formulas, 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 Cartesian point is therefore (√6/2, √6/2, 1). When you use the calculator, you will see decimal approximations of these values, which is normal for a numerical tool.
A practical way to check your own work is to convert in one direction and then convert back. If you start with Cartesian coordinates, convert to spherical, and then convert the result back to Cartesian, you should recover the original point up to small rounding differences. That round-trip check is especially useful when you are learning the formulas or debugging a 3D geometry workflow.
Interpreting the result
After a conversion, think about what the numbers mean geometrically. A larger r means the point is farther from the origin. A small r means the point is close to the origin. If θ is near 0, the point lies close to the positive z-axis. If θ is near π, the point lies close to the negative z-axis. If θ is near π/2, the point is near the xy-plane.
The azimuth φ tells you the direction around the vertical axis. When φ is near 0, the point is aligned with the positive x-direction in the horizontal plane. When φ is near π/2, it points toward positive y. Other values place the point in the remaining quadrants. This is why the atan2 function is so important: it preserves the correct directional information instead of losing the sign pattern of x and y.
It is also normal for more than one spherical description to represent the same location. Adding 2π to φ does not change the point. At the same time, some equivalent descriptions can be written by adjusting angles in ways that preserve the same direction. The calculator reports one consistent numerical representation, which is usually the most useful form for computation.
Applications and comparison with Cartesian coordinates
Cartesian coordinates are often best when motion or geometry is naturally aligned with straight axes. They are common in algebra, computer graphics, and rectangular grids. Spherical coordinates become more natural when a problem depends mainly on distance from a center point or on angular direction. Examples include gravitational fields, electric fields around a point charge, radiation patterns, wave propagation, and surfaces such as spheres and shells.
One reason spherical coordinates are so important in calculus and physics is that they simplify symmetric problems. A sphere of fixed radius is simply written as r = constant, while in Cartesian form it becomes x² + y² + z² = r². Likewise, many integrals over spherical regions become easier to set up in spherical coordinates, even though the volume element changes to r² sin(θ) · dr · dθ · dφ. That extra factor reflects the geometry of how small wedges of space expand as radius and angle change.
This converter is therefore useful not only for direct coordinate changes but also for building intuition. If you are studying multivariable calculus, electromagnetism, mechanics, astronomy, or 3D programming, being able to move comfortably between coordinate systems helps you choose the representation that makes the structure of the problem easiest to see.
Limitations and assumptions
This calculator assumes the standard physics convention described above and expects all angles in radians. If your course or software uses a different naming convention for θ and φ, the numerical values may still describe the same point but the labels will not match. Always compare conventions before concluding that two answers disagree.
There is also an important edge case at the origin. When r = 0, the point is exactly at (0, 0, 0), and the angles are mathematically undefined because every direction points to the same location. The existing calculator logic may display a default numerical result in that situation, but geometrically the angles do not carry meaningful directional information there. In addition, very large or very small values can introduce ordinary floating-point rounding effects, so tiny numerical differences should be expected in extreme cases.
Finally, the tool is intended for coordinate conversion, not symbolic algebra. It returns decimal approximations and does not simplify expressions into exact forms such as π/3 or √2. That makes it fast and practical, but it also means you should interpret the output as a numerical estimate of the exact mathematical result.