Math
Cylindrical Coordinates Calculator
Cylindrical coordinates keep z from the rectangular system and replace x and y with the polar pair (r, θ).
Enter a point in either system and mathcheck converts it, showing r as an exact radical, θ placed in the correct quadrant in both degrees and radians, and exact values like (-2√3, 2, 5) when θ is a special angle. A third mode converts cylindrical to spherical (ρ, θ, φ).
Try 1.7320508 for √3.
Same in both rectangular and cylindrical coordinates.
Try an example
Result
Converted point
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- r
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- θ (degrees)
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- θ (radians)
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- z
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- x
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- y
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More details (2 more)
- ρ (distance to the origin)
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- φ (angle from +z)
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What cylindrical coordinates are
Cylindrical coordinates (r, θ, z) are polar coordinates in the xy-plane with the height z attached. r is the distance from the z-axis, θ is the angle around the z-axis measured counterclockwise from the positive x-axis, and z is the same height as in rectangular coordinates. Surfaces with a natural axis, such as pipes, cans, and helices, have simple equations in this system: r = 2 is a cylinder of radius 2.
| System | Coordinates | Meaning |
|---|---|---|
| Rectangular | (x, y, z) | distances along three perpendicular axes |
| Cylindrical | (r, θ, z) | distance from the z-axis, angle around it, height |
| Spherical | (ρ, θ, φ) | distance from the origin, azimuth angle, angle from +z |
(-1, √3, 3) to cylindrical
r = sqrt(1 + 3) = 2. The reference angle is atan(√3/1) = 60°, and (-1, √3) is in Quadrant II, so θ = 180° - 60° = 120° = 2π/3. z stays 3. Cylindrical form: (2, 120°, 3).
How to use this calculator
- Choose the conversion: rectangular to cylindrical, cylindrical to rectangular, or cylindrical to spherical.
- Enter the coordinates. Angles can be typed in degrees, An angle unit where a full turn is 2π instead of 360 degrees., or as π fractions such as 5pi/6.
- Read the converted point in the first result, with r, θ (degrees and radians), z, x, y, ρ, and φ listed beneath it.
- Open Show the work for the A root expression written with the √ symbol, such as a square root. for r, the reference-angle and One of the four regions the x- and y-axes cut the coordinate plane into. reasoning, and the exact sine and cosine values.
How to read the answer
r is never negative in the rectangular-to-cylindrical result, and θ is reported in [0°, 360°). x and y come out exact (for example -2√3) when θ is a multiple of 30° or 45°, otherwise as four-decimal values. ρ and φ preview the spherical form so you can move between all three systems.
Common mistakes and edge cases
- Including z in r. The cylindrical radius uses only x and y: r = sqrt(x^2 + y^2), not sqrt(x^2 + y^2 + z^2) (that is the spherical ρ).
- Using atan(y/x) without adjusting for the quadrant, which mislabels every point with x < 0.
- Confusing θ (around the z-axis) with the spherical φ (down from the z-axis).
- Typing radians in the degrees box: 2 means 2° there, not 2 radians.
- Expecting exact radicals for every angle. Only multiples of 30° and 45° have exact sine and cosine values.
Worked examples
Quadrant II point to cylindrical
(-1, √3, 3)
Converted point
(r, θ, z) = (2, 120°, 3)
3-4-5 point to cylindrical
(3, 4, -2)
Converted point
(r, θ, z) = (5, 53.1301°, -2)
Point on the z-axis
(0, 0, 7): θ is undefined
Converted point
(r, θ, z) = (0, 0°, 7)
Cylindrical to rectangular, exact
(4, 150°, 3)
Converted point
(x, y, z) = (-2√3, 2, 3)
Cylindrical to rectangular with a π fraction
(2, 3π/4, -1)
Converted point
(x, y, z) = (-√2, √2, -1)
Non-special angle
(5, 2 rad, 1)
Converted point
(x, y, z) = (-2.0807, 4.5465, 1)
Cylindrical to spherical
(3, 90°, 4)
Converted point
(ρ, θ, φ) = (5, 90°, 36.8699°)
Cylindrical to spherical, point below the xy-plane
(1, 45°, -1)
Converted point
(ρ, θ, φ) = (√2, 45°, 135°)
Bad angle text
Converted point
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Frequently asked questions
How do I convert rectangular to cylindrical coordinates?+
Keep z. Compute r = sqrt(x^2 + y^2) and θ from the reference angle atan(|y|/|x|) placed in the correct quadrant. For (-1, √3, 3): r = 2, θ = 120°, z = 3.
How do I convert cylindrical to rectangular coordinates?+
x = r cos θ, y = r sin θ, and z is unchanged. Use exact sine and cosine values at special angles: (4, 150°, 3) becomes (-2√3, 2, 3).
What is the difference between cylindrical and spherical coordinates?+
Cylindrical uses the distance r from the z-axis plus the height z; spherical uses the distance ρ from the origin plus the angle φ from the positive z-axis. They share the azimuth angle θ, and ρ^2 = r^2 + z^2.
Can r be negative in cylindrical coordinates?+
Conventionally r ≥ 0. A negative r is allowed by the formulas and simply points the other way: (-r, θ, z) is the same point as (r, θ + 180°, z). The calculator warns you and shows the standard form.
What is θ for a point on the z-axis?+
Undefined, because x = y = 0 gives no direction. Any θ works, and the calculator shows 0° by convention.
Which equations are simplest in cylindrical coordinates?+
Anything symmetric about the z-axis: the cylinder x^2 + y^2 = 4 is r = 2, the cone z = sqrt(x^2 + y^2) is z = r, and the paraboloid z = x^2 + y^2 is z = r^2.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026