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# Cylinder

A cylinder is one of the basic shapes in 3D geometry. It has two parallel circular bases set at a distance from each other. The two circular bases are joined by a curved surface at a fixed distance from the center.

The circles and their interiors are called the bases. The radius of the cylinder is the radius of the base. The perpendicular segment from the plane of one base to the plane of the other base is called the altitude of the cylinder. The height of the cylinder is the length of the altitude.

The axis of a cylinder is the segment that contains the centers of the two bases. When the axis is perpendicular to the planes of the two bases, you have a right cylinder. If not, you have an oblique cylinder.

## Properties of a cylinder

Each shape has properties that differentiate it from other shapes. Cylinders have the following characteristics:

• The bases are congruent and parallel.
• If the axis forms a right angle with the bases, which are exactly over each other, it is called a "right cylinder".
• It is similar to the prism because all cross sections perpendicular to its altitude are identical.
• If the bases are not directly over each other but still parallel, the axis does not produce a right angle to the bases and it is called an "oblique cylinder".
• If the bases are circular in shape, it is called a "circular cylinder".
• If the bases are an elliptical shape, it is called an "elliptical cylinder".

## Measuring the surface area of a cylinder

In order to find the surface area of a cylinder, we need to consider its 3 outward faces, the two bases, and the tube that connects them.

Since the base of a cylinder is a circle, its area is πr2 where r is the radius of the base of the cylinder.

We also need to consider the tube part, which we call the lateral surface area. Note that you can unroll the tube into a rectangle with height h, and base $2\pi \left(r\right)$ (the circumference of the circular base). This gives us an area of $\left(\mathrm{LSA}\right)=2\pi \left(r\right)\left(h\right)$ .

This means that the total area is the two circles plus the tube, or:

${\mathrm{SA}}_{c}=2\pi {r}^{2}+2\pi rh$

Example 1:

Find the lateral surface area of a cylinder with a base radius of 3 inches and a height of 9 inches.

$\mathrm{LSA}=2\pi \left(3\right)\left(9\right)$

$=54\pi {\mathrm{in}}^{2}$

Approximately $169.56{\mathrm{in}}^{2}$

Example 2:

Find the total surface area of a cylinder with a base radius of 5 inches and a height of 7 inches.

$\mathrm{SA}{C}_{}=2\pi \left(5\right)\left(7\right)+2\pi {\left(5\right)}^{2}$

$=70\pi +50\pi$

$=120\pi {\mathrm{in}}^{2}$

Approximately $376.8{\mathrm{in}}^{2}$

## Finding the volume of a cylinder

If we wanted to find the volume of a cylinder, we must multiply the area of the base by the height of the cylinder. The formula is:

$V=\pi {r}^{2}h$

Example 3

Find the volume of a cylinder with a radius of 3 cm and a height of 10 cm.

$V=\pi {\left(3\right)}^{2}\left(10\right)$

$V=90\pi {\mathrm{cm}}^{3}$

Approximately $282.6{\mathrm{cm}}^{3}$