5
53.130102
0.927295
5
3
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5
5
53.130102
0.927295
5
3
4
5
The Cylindrical Coordinates Calculator converts between Cartesian coordinates $$(x, y, z)$$ and cylindrical coordinates $$(r, \theta, z)$$. Cylindrical coordinates extend the 2D polar system into three dimensions by adding a height component $$z$$, making them ideal for problems with axial symmetry such as pipes, cylinders, solenoids, and rotating machinery.
The conversion from Cartesian to cylindrical is:
$$r = \sqrt{x^2 + y^2}$$
$$\theta = \text{atan2}(y, x)$$
$$z = z$$
The conversion from cylindrical to Cartesian is:
$$x = r\cos\theta$$
$$y = r\sin\theta$$
$$z = z$$
Note that the $$z$$-coordinate remains unchanged in both directions. The transformation only affects the $$xy$$-plane, converting between rectangular and polar representations.
The volume element in cylindrical coordinates is:
$$dV = r\,dr\,d\theta\,dz$$
The extra factor of $$r$$ (the Jacobian) accounts for the non-uniform spacing of the coordinate grid. Grid cells farther from the $$z$$-axis cover more area than those close to it.
Cylindrical coordinates are the natural choice for a wide variety of physical problems. In electromagnetism, the magnetic field of an infinite straight wire, a solenoid, or a coaxial cable is most simply expressed in cylindrical form. For a long straight wire carrying current $$I$$, the magnetic field at distance $$r$$ is:
$$B = \frac{\mu_0 I}{2\pi r}$$
In fluid dynamics, pipe flow (Poiseuille flow) has a parabolic velocity profile that depends only on $$r$$:
$$v(r) = v_{\max}\left(1 - \frac{r^2}{R^2}\right)$$
In heat transfer, radial heat conduction through a cylindrical wall uses the cylindrical form of the heat equation. In structural engineering, stress analysis of cylindrical pressure vessels uses cylindrical coordinates to exploit the symmetry.
The Laplacian operator in cylindrical coordinates is:
$$\nabla^2 f = \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial f}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 f}{\partial\theta^2} + \frac{\partial^2 f}{\partial z^2}$$
This is fundamental for solving Laplace's equation, the wave equation, and the diffusion equation in cylindrical geometries. Separation of variables in cylindrical coordinates yields Bessel functions, which arise in vibration analysis, electromagnetic waveguides, and quantum mechanics.
Select the conversion direction. For Cartesian to Cylindrical, enter $$(x, y, z)$$. The calculator computes $$(r, \theta, z)$$ where $$r$$ and $$\theta$$ come from the $$xy$$-plane projection. For Cylindrical to Cartesian, enter $$(r, \theta, z)$$ and receive the $$(x, y, z)$$ equivalents.
The radial distance $$r$$ measures how far the point is from the $$z$$-axis (not the origin). The angle $$\theta$$ is measured in the $$xy$$-plane from the positive $$x$$-axis. The height $$z$$ is the same in both systems. Points on the $$z$$-axis have $$r = 0$$ and $$\theta$$ is undefined.
Inputs
Results
The point (3, 4, 5) is at radial distance 5 from the z-axis, angle ≈ 53.13° in the xy-plane, and height z = 5.
Inputs
Results
At r = 10 and θ = 30°: x = 10cos(30°) ≈ 8.66, y = 10sin(30°) = 5, z = 7.
Cylindrical coordinates are the 3D extension of 2D polar coordinates. They add a $$z$$-axis (height) to the polar $$(r, \theta)$$ pair. The $$r$$ and $$\theta$$ components work identically to polar coordinates in the $$xy$$-plane.
The Jacobian determinant is $$r$$. This means the volume element transforms as $$dx\,dy\,dz = r\,dr\,d\theta\,dz$$. The factor $$r$$ reflects that coordinate cells at larger radii cover more volume.
Use cylindrical coordinates when the problem has symmetry about an axis — pipes, cylinders, solenoids, rotating shafts, circular antennas. The symmetry reduces the number of independent variables, simplifying equations.
Bessel functions are solutions to Bessel's differential equation, which arises when separating variables in the Laplacian in cylindrical coordinates. They describe radial modes of vibration in drums, electromagnetic fields in waveguides, and heat distribution in cylinders.
The gradient is $$\nabla f = \frac{\partial f}{\partial r}\hat{r} + \frac{1}{r}\frac{\partial f}{\partial\theta}\hat{\theta} + \frac{\partial f}{\partial z}\hat{z}$$. The $$1/r$$ factor in the $$\theta$$-component accounts for the angular nature of the coordinate.
By standard convention, $$r \geq 0$$. Negative $$r$$ values are sometimes used in polar plots where $$(-r, \theta)$$ means $$(r, \theta + \pi)$$, but this is uncommon in cylindrical coordinates for physical applications.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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