Cylindrical Coordinates Calculator

A cylindrical coordinates calculator converts Cartesian coordinates to a unit of its equivalent value in cylindrical coordinates and vice versa. It does this by taking a unit's rectangular (or cartesian) coordinates and converting them to their equal value in cylindrical coordinates. The Mathematics Master online Cylindrical Coordinates calculator is simple and incredibly beneficial to its users.

Cartesian (x, y, z) to cylindrical (ρ, θ, z)

Cylindrical Coordinates

In three-dimensional space, cylindrical coordinates are a logical extension of polar coordinates. A collection of three cylindrical coordinates can be used to identify a point in the cylindrical coordinate system. We can use cartesian and polar coordinates to specify a point’s location in two dimensions.

An additional z coordinate is introduced when the polar coordinates are expanded to a three-dimensional plane. Together, these three measurements create cylindrical coordinates.

Change From Rectangular to Cylindrical Coordinates

Recall that a point P in three dimensions is represented by the ordered triple (r, θ, z) in the cylindrical coordinate system. Where r and θ are the polar coordinates of the projection of point P onto the XY-plane and z is the directed distance from the XY-plane to P.

Use the following formula to convert rectangular coordinates to cylindrical coordinates.

\( r^2 = x^2 + y^2 \)

\( tan(θ) = \dfrac{y}{x} \)

\( z = z \)

Example: Rectangular to Cylindrical Coordinates

Let’s take an example with rectangular coordinates (3, -3, -7) to find cylindrical coordinates.

Substitute the specified ordered triple into the formulae shown above to convert from rectangular to cylindrical coordinates. Remember that the coordinates supplied can be understood as x = 3, y= -3, and r = -7.

\( r = \sqrt{x^2 + y^2} \)

\( r = \sqrt{(3)^2 + (-3)^2} \)

\( r = 3\sqrt{2} \)

\( tan(θ) = \dfrac{y}{x} \)

\( tan(θ) = \dfrac{-3}{3} \)

\( tan (θ) = -1 \)

\( θ = tan-1 (-1) \)

\( θ = \dfrac{3π}{4} + πn \)

\( z = z \)

\( z = -7 \)

We can write the converted cylindrical coordinates (3, -3, -7) in several forms, such as (\(3\sqrt{2}\), \(\frac{3π}{4}\), -7) and (\(3\sqrt{2}\), \(\frac{7π}{4}\), -7).

Key Points on Cylindrical Coordinates

A plane’s radial distance, azimuthal angle, and height are used to locate a point in the cylindrical coordinate system. These coordinates are ordered triples.

The symbol for cylindrical coordinates is (r, θ, z).

We can transform spherical and cylindrical coordinates into cartesian coordinates and vice versa.


Are polar and cylindrical coordinates the same?

Simple three-dimensional extensions of the two-dimensional polar coordinates are known as cylindrical coordinates. Remember that we can use polar coordinates (r,θ) to define a point’s location in a plane. For example, the distance of the point from the origin is represented by the polar coordinate r.

What is the radial distance?

It is the distance in Euclidean geometry between the point in three-dimensional space and the origin, O (0, 0).

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