# Cylindrical Coordinates: Rectangular to Cylindrical Coordinates Conversion and Vice Versa

*Ray is a licensed engineer in the Philippines. He loves to write about mathematics and civil engineering.*

In the cylindrical coordinate system, a point P in three-dimensional space is represented by the ordered triple (r, θ, z), where r and θ are polar coordinates of P's projection onto the XY-plane and z is the directed distance from the XY-plane to P.

In three dimensions, two coordinate systems are similar to polar coordinates and give convenient descriptions of some commonly occurring surfaces and solids.

If not familiar, the polar coordinate system gives a convenient description of individual curves and regions. Recall the connection between polar coordinates and cartesian coordinates.

**Change From Rectangular to Cylindrical Coordinates and Vice Versa**

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

To convert from rectangular to cylindrical coordinates, use the formulas presented below.

r^{2} = x^{2} + y^{2}

tan(θ) = y/x

z = z

To convert from cylindrical to rectangular coordinates, use the following equations.

x = r cos(θ)

y = r sin (θ)

z = z

Cylindrical coordinates are useful in problems that involve symmetry about an axis, and the z-axis is chosen to coincide with this axis of symmetry. For instance, the circular cylinder axis with Cartesian equation x^{2} + y^{2} = c^{2} is the z-axis. In cylindrical coordinates, the cylinder has the straightforward equation r = c. It is the reason for the name "cylindrical" coordinates.

## Example 1: Conversion Between Cylindrical and Rectangular Coordinates

Plot the point with cylindrical coordinates (2, 2π/3, 1) and find its rectangular coordinates.

**Solution**

The given problem is a conversion from cylindrical coordinates to rectangular coordinates. First, plot the given cylindrical coordinates or the triple points in the 3D-plane as shown in the figure below. Next, substitute the given values in the mentioned formulas for cylindrical to rectangular coordinates.

Recall that the dimensional space is represented by the ordered triple (r, θ, z) whereas r = 2, θ = 2π/3, and z = 1.

x = r cos(θ)

x = (2) cos (2π/3)

x = (2) (-1/2)

x = -1

y = r sin (θ)

y = (2) sin (2π/3)

y = √3

z = z

z = 1

**Answer**

The rectangular coordinates of (2, 2π/3, 1) are (-1, √3, 1).

## Example 2: Rectangular to Cylindrical Coordinates

Find cylindrical coordinates of the point with rectangular coordinates (3, -3, -7).

**Solution**

When converting rectangular coordinates to cylindrical coordinates, simply substitute the given ordered triple to the introduced equations above. Recall that the given coordinates can be interpreted as x = 3, y= -3, and r = -7.

r = √x^{2} + y^{2}

r = √(3)^{2} + (-3)^{2}

r = 3√2

tan (θ) = y/x

tan (θ) = -3/3

tan (θ) = -1

θ = tan^{-1} (-1)

θ = (3π/4) + πn

z = z

z = -7

**Answer**

The converted cylindrical coordinates of the given rectangular coordinates (3, -3, -7) can be expressed in multiple forms such as (3√2, 3π/4, -7) and (3√2, 7π/4, -7), and many more. As with polar coordinates, there are infinitely many solutions.

## Example 3: Identifying the Surface Given an Equation

Describe the surface whose equation in cylindrical coordinates is z = r.

**Solution**

The given equation z = r interprets that the value of z or height of each point on the surface is the same as r, the distance from the point to the z-axis. Since θ does not appear, it can vary. Any horizontal trace in the plane z = k (k > 0) is a circle of radius k. These traces suggest that the surface is a cone. This prediction can be confirmed by converting the equation into rectangular coordinates. From the first equation, we have the following.

z^{2} = r^{2}

z^{2} = x^{2} + y^{2}

**Answer**

We recognize the equation z^{2} = x^{2} + y^{2} as a circular cone whose axis is the z-axis.

## Example 4: Cylindrical Equation for an Ellipsoid

Find an equation in cylindrical coordinates for the ellipsoid 4x^{2} + 4y^{2} + z^{2} = 1.

**Solution**

Substitute r^{2} to x^{2} + y^{2} and simplify the given equation of ellipsoid in cylindrical coordinates.

4x^{2} + 4y^{2} + z^{2} = 1

z^{2} = 1 - 4x^{2} + 4y^{2}

z^{2} = 1 – 4(x^{2} + y^{2})

z^{2} = 1 – 4r^{2}

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**Answer**

The equation of the ellipsoid in cylindrical coordinates is z^{2} = 1 – 4r^{2}.

## Example 5: Identifying the Surface Given an Equation

Identify the surface for each of the following equations.

a. r = 10

b. r^{2} + z^{2} = 81

**Solution**

In two-dimensional space, we can infer that the first given equation is a circle of radius 10. Since we are now in three dimensions and there is no z in the equation, it can vary freely.

Therefore, for any given z, we have a circle of radius 5 with its center lying on the z-axis.

For the second equation, substitute r^{2} = x^{2} + y^{2} into the equation r^{2} + z^{2} = 81 to express the rectangular form of the equation.

r^{2} + z^{2} = 81

x^{2} + y^{2} + z^{2} = 81

**Answer**

a. The equation is a cylinder of radius 10 centered on the z-axis.

b. The equation is a sphere centered at the origin with radius 9.

## Example 6: Converting Cylindrical to Rectangular Coordinates

Plot the point with the cylindrical coordinates (5, 2π/4, -3) and express its location in rectangular coordinates.

**Solution**

First, plot the coordinates in the three-dimensional space to visualize properly. Then, apply the equations listed above for the conversion from cylindrical coordinates to rectangular coordinates. Given (5, 2π/4, -3), the rectangular coordinates shall be:

x = r cos(θ)

x = (5) cos (2π/4)

x = (5) (0)

x = 0

y = r sin (θ)

y = (5) sin (2π/4)

y = (5) (1)

y = 5

z = z

z = -3

**Answer**

The point with cylindrical coordinates (5, 2π/4, -3) has rectangular coordinates (0, -1, -3).

## Example 7: Rectangular to Cylindrical Coordinates

Convert the rectangular coordinates (1, -2, 6) to cylindrical coordinates.

**Solution**

To translate from rectangular to cylindrical coordinates, simply substitute to the right equations. Recall that the given coordinates can be interpreted as x = 1, y= -2, and r = 6.

r = √x^{2} + y^{2}

r = √(1)^{2} + (-2)^{2}

r = √5

tan (θ) = y/x

tan (θ) = -2/1

tan (θ) = -2

θ = tan^{-1} (-2)

θ = 2.0344 + πn

z = z

z = 6

**Answer**

The converted cylindrical coordinates of the given rectangular coordinates (1, -2, 6) can be expressed in multiple forms such as (3√2, 2.034, -7) and many more. As with polar coordinates, there are infinitely many solutions.

## Example 8: Identifying the Surfaces in the Cylindrical Coordinate System

Describe the surfaces with the given cylindrical equations.

a. θ = π/2

b. r^{2} + z^{2} = 9

**Solution**

When the angle theta is held constant whole r and z are allowed to vary, the result is a half-plane.

First, substitute r^{2} = x^{2} + y^{2} into the given equation r^{2} + z^{2} = 9 in order to arrive to its rectangular form.

r^{2} = x^{2} + y^{2}

r^{2} + z^{2} = 9

x^{2} + y^{2} + z^{2} = 9

The resulting equation is a sphere centered at the origin with radius 3.

**Answer**

Therefore, θ = π/2 is a half-plane and r^{2} + z^{2} = 9 is a sphere.

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*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*