Cylindrical coordinates have the form (*r, θ, z*), where *r* is the distance in the *xy* plane, θ is the angle formed with respect to the *x-axis*, and z is the vertical component in the z-axis. Similar to polar coordinates, we can relate cylindrical coordinates to Cartesian coordinates by using a right triangle and trigonometry. We use cosine to find the *x* component and sine to find the *y* component. The component in *z* stays the same.

Here, we will look at the formulas that we can apply to transform from cylindrical to Cartesian coordinates. Then, we will use these formulas to solve some practice exercises.

##### TRIGONOMETRY

**Relevant for**…

Learning to transform from cylindrical to Cartesian coordinates.

##### TRIGONOMETRY

**Relevant for**…

Learning to transform from cylindrical to Cartesian coordinates.

## How to transform from cylindrical coordinates to Cartesian coordinates?

Three-dimensional Cartesian coordinates are represented in the form $latex (x, y, z)$. An alternate three-dimensional coordinate system is the cylindrical coordinate system. The cylindrical coordinate system is a three-dimensional extension of the polar coordinate system.

The polar coordinates are extended in the third dimension similar to how we extended the Cartesian coordinates, that is, we simply add *z* as the third dimension.

Then, the cylindrical coordinates have the form $latex (r, θ, z)$, where, r is the distance from the origin to the position of the point in the xy plane, θ is the angle with respect to the x-axis, and z is the coordinate in the z-axis.

The transformations for *x* and *y* are the same as those used in polar coordinates. To find the *x* component, we use the cosine function, and to find the *y* component, we use the sine function.

Also, the *z* component of the cylindrical coordinates is equal to the *z* component of the Cartesian coordinates.

$latex x=r~\cos(\theta)$ $latex y=r~\sin(\theta)$ $latex z=z~~~~~$ |

## Cylindrical to Cartesian coordinates – Examples with answers

The formulas for the transformation of cylindrical coordinates to Cartesian coordinates are used to solve the following examples. Each example has its respective solution, but it is recommended that you try to solve the problems yourself before looking at the answer.

**EXAMPLE 1**

We have the point (3, 30°, 6) in cylindrical coordinates. What is its equivalence in Cartesian coordinates?

##### Solution

We start with the values $latex r=3,~\theta=30^{\circ},~z=6$. Using these values and the formulas seen above, we have to find the values of *x* and *y*. Therefore, the value of *x* is:

$latex x=r~\cos(\theta)$

$latex x=3~\cos(30)$

$latex x=2.6$

The value of *y* is:

$latex y=r~\sin(\theta)$

$latex y=3~\sin(30)$

$latex y=1.5$

We know that the *z* component remains the same, so the coordinates of the point are (2.6, 1.5, 6).

**EXAMPLE 2**

What are the Cartesian coordinates of the point (5, 45°, -4)?

##### Solution

We use the values $latex r=5,~\theta=45^{\circ},~z=-4$ to find the different components of the Cartesian coordinates. We start with the value of *x*:

$latex x=r~\cos(\theta)$

$latex x=5~\cos(45)$

$latex x=3.54$

Now, we find the value of *y*:

$latex y=r~\sin(\theta)$

$latex y=5~\sin(45)$

$latex y=3.54$

The *z* component is the same, so the Cartesian coordinates of the point are (3.54, 3.54, -4).

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**EXAMPLE 3**

We have a point with cylindrical coordinates (6, 120°, 7). What are the Cartesian coordinates of this point?

##### Solution

We find the Cartesian coordinates using the values $latex r=6,~\theta=120^{\circ},~z=7$ along with the formulas seen above. Therefore, the value of *x* is:

$latex x=r~\cos(\theta)$

$latex x=6~\cos(120)$

$latex x=-3$

The value of *y* is:

$latex y=r~\sin(\theta)$

$latex y=6~\sin(120)$

$latex y=5.2$

The *z* component is the same, so the coordinates of the point are (-3, 5.2, 7).

**EXAMPLE 4**

We have the point (12, 90°, 8) in cylindrical coordinates. What is its equivalence in Cartesian coordinates?

##### Solution

We have the values $latex r=12,~\theta=90^{\circ},~z=8$. We find the value of *x* as follows:

$latex x=r~\cos(\theta)$

$latex x=12~\cos(90)$

$latex x=0$

We find the value of *y* as follows:

$latex y=r~\sin(\theta)$

$latex y=12~\sin(90)$

$latex y=12$

We know that the *z* component remains the same, so the coordinates of the point are (0, 12, 8).

## Cylindrical to Cartesian coordinates – Practice problems

Use the formulas seen above to solve the following problems and transform the cylindrical coordinates to Cartesian coordinates. Select an answer and check it to see if you got the correct answer.

## See also

Interested in learning more about cylindrical coordinates? Take a look at these pages:

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