Cylindrical Coordinates Calculator - (x, y, z) to (r, theta, z) and Back

A cylindrical coordinates calculator turns a 3D point written as (x, y, z), (r, theta, z), or (rho, phi, theta) into the same point in one of the other two systems. It applies the conversion pairs r = sqrt(x^2 + y^2), theta = atan2(y, x), x = r cos(theta), y = r sin(theta), rho = sqrt(r^2 + z^2), phi = atan2(r, z), r = rho sin(phi), and z = rho cos(phi).

• • Convert a Cartesian point to cylindrical form: Type (x, y, z) = (3, 4, 5) and read (r, theta, z) = (5, 53.13 deg, 5).
• • Convert a cylindrical point to Cartesian for code: Type (r, theta, z) = (5, 60 deg, 7) and read (x, y, z) = (2.5, 4.3301, 7).
• • Move from cylindrical to spherical for triple integrals: Type (r, theta, z) = (4, 53.13 deg, 3) and read (rho, phi, theta) = (5, 53.13 deg, 53.13 deg).
• • Recover cylindrical from a spherical measurement: Type (rho, phi, theta) = (5, 53.13 deg, 45 deg) and read (r, theta, z) = (4, 45 deg, 3).

Cartesian, cylindrical, and spherical coordinates are three different ruler sets for the same 3D space. The conversion is exact: (3, 4, 5) in Cartesian and (5, 53.13 deg, 5) in cylindrical describe the same position.

For a 2D problem on the xy-plane without a z height, Cartesian to Polar Calculator does the same conversion for the pair (x, y) to (r, theta) with no extra z input.

The calculator reads the chosen direction, treats the three input numbers as the source-system coordinates, and applies the matching trigonometric formulas. The result panel shows the same five values, and the labels switch to match the current direction.

• x, y, z: Cartesian coordinates. x and y set the radial distance and azimuthal angle; z is the height.
• r: Radial distance in the xy-plane, equal to sqrt(x^2 + y^2). Always non-negative.
• theta: Azimuthal angle in degrees and radians, measured counterclockwise from the positive x-axis. atan2 picks the correct quadrant.
• rho: Spherical radial distance from the origin, equal to sqrt(x^2 + y^2 + z^2). Always non-negative.
• phi: Spherical polar angle from the positive z-axis downward: 0 on the positive z-axis, 90 deg on the xy-plane, 180 deg on the negative z-axis.

The five result rows always have the same ids and precision, but their meaning switches with the direction selector. For cartesian-to-cylindrical the rows are the radial distance, theta in degrees, theta in radians, the input z echoed back, and the quadrant label. For cylindrical-to-cartesian the same rows become x, y, z, the input theta, and a reference label.

All three conversion pairs are exact, so the calculator does not introduce rounding beyond the four-decimal display precision in degrees and the six-decimal display precision in radians. The only place where the answer is symbolic is the origin and the z-axis, where theta is mathematically undefined.

According to Wolfram MathWorld, the conversion between Cartesian (x, y, z) and cylindrical (r, theta, z) uses x = r cos(theta), y = r sin(theta), r = sqrt(x^2 + y^2), and theta = atan2(y, x), with the z component unchanged in both directions

According to Wikipedia (Atan2), atan2(y, x) returns the angle whose tangent is y/x with the correct sign for all four quadrants, in the open interval (-pi, pi]

When the conversion is followed by a distance or norm calculation in 3D, 3D Distance Calculator computes the same sqrt(x^2 + y^2 + z^2) value from the same input triple.

Four ideas cover every cylindrical-coordinates conversion: the three coordinate systems, the azimuthal angle, the right-triangle interpretation of rho and phi, and how atan2 handles the four quadrants.

These four ideas cover every conversion. Once you know that theta is the same in cylindrical and spherical, and that the (r, z) pair forms a right triangle with hypotenuse rho, every other conversion is just a rearrangement of the same formulas.

Since atan2 is the two-argument generalization of arctan, Arctan Calculator handles the single-argument case when the point sits on the x-axis or y-axis.

Five short steps take any 3D point from the source coordinate system to the destination system.

1. 1 Pick the conversion direction: Open the Conversion direction menu and select the pair of systems.
2. 2 Enter the three source coordinates: Type the three values for the source system. For Cartesian, type x, y, and z. For cylindrical, type r, theta in degrees, and z. For spherical, type rho, phi in degrees, and theta in degrees.
3. 3 Read the destination coordinates: The result panel shows the destination coordinates with four decimals for distances and degrees, and six decimals for radians.
4. 4 Switch angle units between degrees and radians: Both theta and phi are reported in degrees and radians at the same time.
5. 5 Use the reference label to confirm the region: The bottom row names the quadrant, the axis, or the system label.

A satellite is 60 units from the origin on the xy-plane, 25 units above it, at azimuth 35 degrees. Set direction to Cylindrical to Spherical, enter x=60, y=25, z=35. The result reads rho=65, phi=67.38 degrees, and theta=35 degrees.

When the conversion is the first step before a norm or magnitude calculation, Vector Magnitude Calculator returns just sqrt(x^2 + y^2 + z^2) from the same three input values.

The cylindrical coordinates calculator handles the three conversion pairs, the degree/radian output, and the indeterminate cases that usually trip up hand calculations.

• • All three systems in one place: Cartesian, cylindrical, and spherical conversions live in the same panel.
• • Both angle units in one result row: Degrees and radians appear in the same direction at the same time, so the value can be pasted into a degree-based or radian-based formula.
• • Atan2 picks the correct quadrant automatically: The two-argument arctangent handles every quadrant and the two axes.
• • Origin and z-axis handled explicitly: When the point is at the origin or on the z-axis, the azimuthal angle is mathematically undefined. The calculator reports 'Undefined' instead of a silent NaN.
• • Pairs with related 2D and 3D tools: The same r and theta feed into the cartesian to polar calculator, and the same x, y, z feed into the 3d distance calculator and the vector magnitude calculator.

Most undergraduate physics, engineering, and math courses use at least two of the three coordinate systems in the same problem set. Having one place to flip between them keeps the math right.

When the next step is a perpendicular vector or a parallelogram area, Cross Product Calculator takes two (x, y, z) triples and returns the cross product.

Three things change the answer you should expect, plus two practical caveats about how cylindrical coordinates are usually defined.

• • The cylindrical form of a non-origin point is not unique: (r, theta, z) and (r, theta + 360 deg, z) describe the same point. The calculator reports the principal value of theta in (-180 deg, 180 deg].
• • The phi in spherical coordinates is the polar angle from the positive z-axis, not the geographic latitude. If a problem uses latitude, the conversion is phi = 90 deg - latitude.

When the result feeds into a triple integral, the Jacobian in cylindrical coordinates is r and in spherical coordinates is rho^2 * sin(phi).

According to Wikipedia (Cylindrical coordinate system), the cylindrical system extends the polar system to 3D by adding a third z coordinate, and relates to spherical coordinates by rho = sqrt(r^2 + z^2) and phi = atan2(r, z)

When the surrounding problem also needs latitude and longitude or DMS notation, Coordinates Converter covers the geographic coordinate conversions that sit beside cylindrical and spherical.