How this instrument works
Cylindrical coordinates locate a point in space by keeping the ordinary two-dimensional polar description in the flat plane and stacking a third, untouched coordinate on top of it. The radius r measures the straight-line distance out from a central axis, the angle θ measures how far the point has swept around that axis, and z measures plain vertical height, exactly as it would in Cartesian coordinates. Converting to Cartesian x and y is just resolving that circular sweep into horizontal and vertical parts using the two legs of a right triangle whose hypotenuse is r: x = r·cos θ gives the run, and y = r·sin θ gives the rise.
The shape of the formula comes straight from the unit circle. Walk θ radians counterclockwise from the reference direction on a circle of radius r, and the point reached sits at horizontal position r·cos θ and vertical position r·sin θ by the definitions of sine and cosine alone — z never enters that step, because rotation and height are independent directions and the axis itself never moves. That independence is exactly why cylindrical coordinates exist as their own system: a drill bit, a length of pipe, or the magnetic field around a straight wire all have circular symmetry about one line, so how far out, how far around, and how far along are the three quantities that actually matter, and the third never interacts with the other two.
One case collapses the description: on the axis itself r = 0, and every angle θ names the identical point, because there is no radius left to point in any particular direction. Off the axis, θ still repeats every full turn — adding 2π radians returns exactly the same x and y — so a conversion like this one only ever needs a single representative angle per point, never every equivalent one. z needs no formula of its own; it passes through unchanged, because a cylindrical system and a Cartesian one already agree on what height means.
- Enter the point's distance from the central axis into r (radial distance) — always zero or positive.
- Enter the rotation swept around that axis into θ (angle), switching the unit toggle between degrees, radians, and turns as needed.
- Enter the height along the axis into z (height); this value passes straight through to the output.
- Read x and y for the point's position in the flat plane, computed from r (radial distance) and θ (angle) alone.
- Compare the output z against what you typed into z (height) — the two always match, since height never depends on rotation.
Worked example — a point at r = 5, θ = 60°, z = 3
Take a point five units out from the axis, sixty degrees around from the reference direction, and three units up: r = 5, θ = 60° (which is π⁄3, or 1.0471975511965976 radians), z = 3. Resolve the sweep into flat coordinates: x = 5 × cos(60°) = 5 × 0.5 = 2.5, and y = 5 × sin(60°) = 5 × 0.8660254037844387 = 4.330127018922193.
The height carries over exactly, so the Cartesian point sits at (2.5, 4.330127018922193, 3). A quick check confirms the radius survived the trip: √(2.5² + 4.330127018922193²) = √(6.25 + 18.75) = √25 = 5, matching the r that went in — and 60° is exactly one-sixth of a full turn, which is why the angle came out to a clean fraction of π rather than an untidy decimal.
Questions
What are cylindrical coordinates used for?
They describe points in situations with a natural axis of symmetry: the electric field around a charged wire, the velocity profile of fluid moving through a pipe, or a point on a rotating machine part. In each case, distance from the axis and rotation around it are the physically meaningful quantities, while height along the axis behaves like an ordinary Cartesian coordinate throughout.
How is this different from converting Cartesian to polar coordinates?
Cartesian-to-polar conversion works entirely in a flat, two-dimensional plane and has no third coordinate at all. Cylindrical coordinates take that same polar pair (r, θ) and lift it into three dimensions by adding an untouched height z, so the x = r cos θ and y = r sin θ formulas are identical — the only difference is a third coordinate riding along unchanged.
Why does z not appear in the formulas for x and y?
Because rotation around the axis and movement along the axis are independent directions — sweeping through an angle θ never changes how high up the point sits, and moving up or down never changes how far around the point has swept. That independence is why cylindrical coordinates split into a 2D polar part and a separate linear part rather than mixing all three into one calculation.
What happens when r is 0?
Every point on the axis itself has r = 0, and at that radius θ becomes meaningless — infinitely many values of θ all describe the identical point, since there is no direction left to measure once the distance from the axis has vanished. x and y both come out as 0 regardless of whatever θ was entered.
Do I need to enter θ in radians, or can I use degrees?
Either works, as long as the unit toggle matches what you type; this sheet converts degrees or turns to radians internally before applying cos and sin, since those functions are defined on radians. Sixty degrees and π⁄3 radians describe the identical angle and produce the identical x and y.
Is this the same coordinate system used in spherical coordinates?
No — spherical coordinates replace both r and z with a single distance from the origin plus a second angle measured from the vertical axis, folding height into that distance-and-angle pair rather than keeping it separate. Cylindrical coordinates keep height as its own untouched axis, which is what makes them the natural choice for pipes, wires, and anything else built around one straight line rather than one central point.