The surface area of a cylinder represents the sum of the areas of its faces and is a two-dimensional measure. On the other hand, volume is a measure of the three-dimensional space occupied by the cylinder. **We can calculate the surface area of a cylinder using the formula A=2πr(r+h) and we can calculate its volume using the formula V=πr²h, where r is the radius and h is the height of the cylinder.**

In this article, we will learn about the surface area and volume of a cylinder. We will explore the different formulas that we can use to calculate these measurements, and we will apply them to solve some problems.

## How to find the surface area of a cylinder?

To calculate the surface area of a cylinder, we have to add the areas of all the faces of the cylinder. Then, we can use the following formulas:

$latex A_{s}=2\pi {{r}^2}+2\pi r h$ or $latex A_{s}=2\pi r(r+h)$ |

where, *r* is the length of the radius and *h* is the height of the cylinder.

### Proof of the formula for the surface area of a cylinder

To prove the formula for the surface area of a cylinder, we consider the total area of a cylinder to be made up of the following parts:

- Bases area
- Curved surface area

#### Area of the cylinder bases

The cylinder has two circular bases, so we can calculate its area using the formula for the area of a circle. Since we have two circular bases, we multiply the formula for the area of a circle by 2:

$latex \text{Area}_{\text{bases}}=2\pi{{r}^2}$

#### Area of the curved surface of the cylinder

The lateral surface area can be calculated using the diagram shown below. If we stretch this surface, we form a rectangle with a height of *h* and a base that is equal to the circumference of the circular bases, that is, 2*πr*.

Therefore, we can find this area with the following formula:

$latex \text{Area}_{\text{lateral}}=2\pi r h$

#### Total cylinder surface area

To find the total surface area of the cylinder, we have to add the area of the circular bases and the area of the lateral surface:

$latex A_{s}=2\pi {{r}^2}+2\pi r h$

or

$latex A_{s}=2\pi r(r+h)$

## How to find the volume of a cylinder?

We can calculate the volume of a cylinder by multiplying the area of the base by its height:

Volume = Base × Height

Since the bases of a cylinder are circular, the area of the base is equal to *πr*², where *r* is the radius. Therefore, the formula for the volume of a cylinder is:

$latex V=\pi {{r}^2}\times h$ |

where, *r* is the length of the radius of the cylinder and *h* is the length of its height.

### Calculate the volume of a cylinder using the diameter

To calculate the volume of a cylinder using the diameter, we can use two different methods. The first method consists in dividing the diameter by 2 to get the length of the radius and applying the formula for volume shown above.

Alternatively, we can find a formula for the volume of a cylinder in terms of the diameter by substituting the expression *r*=*d*/2:

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

$latex V=\pi {{(\frac{d}{2})}^2}\times h$

$latex V=\pi \frac{{{d}^2}}{4}\times h$

$latex V= \frac{\pi{{d}^2}}{4}\times h$ |

where, *d* is the length of the diameter.

### How to find the volume of a hollow cylinder

We can calculate the volume of a hollow cylinder by subtracting the volume of the hollow part from the total volume of the cylinder. This can be achieved using the following formula:

$latex V=\pi h({{r_{1}}^2}-{{r_{2}}^2})$ |

where, $latex r_{1}$ is the radius of the cylinder, $latex r_{2}$ is the radius of the hollow part, and *h* is the height of the cylinder.

This formula is equivalent to calculating the volume of an entire cylinder and then extracting the volume of the inner cylinder that was removed.

## Area and volume of a cylinder – Examples with answers

The formulas for the surface area and volume of a cylinder are used to solve the following examples. Try to solve the problems yourself before looking at the solution.

**EXAMPLE 1**

Find the surface area of a cylinder that has a radius of 5 yards and a height of 8 yards.

##### Solution

We have the following lengths:

- Radius, $latex r=5$
- Height, $latex h=8$

Using these lengths in the formula for the surface area, we have:

$latex A_{s}=2\pi r(r+h)$

$latex A_{s}=2\pi (5)(5+8)$

$latex A_{s}=2\pi (5)(13)$

$latex A_{s}=408.4$

The surface area is equal to 408.4 yd².

**EXAMPLE **2

**EXAMPLE**

Find the volume of a cylinder that has a radius of 5 inches and a height of 10 inches.

##### Solution

We have the following:

- Radius, $latex r=5$
- Height, $latex h=10$

Applying the formula for the volume with the given lengths, we have:

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

$latex V=\pi {{(5)}^2}\times 10$

$latex V=\pi (25)\times 10$

$latex V=785.4$

The volume is equal to 785.4 in³.

**EXAMPLE **3

**EXAMPLE**

What is the surface area of a cylinder that has a height of 7 feet and a radius of 6 feet?

##### Solution

We have the following:

- Radius, $latex r=6$
- Height, $latex h=7$

Using these values in the formula for the surface area, we have:

$latex A_{s}=2\pi r(r+h)$

$latex A_{s}=2\pi (6)(6+7)$

$latex A_{s}=2\pi (6)(13)$

$latex A_{s}=490.1$

The surface area is equal to 490.1 ft².

**EXAMPLE **4

**EXAMPLE**

Find the volume of a cylinder that has a radius of 6 inches and a height of 8 inches.

##### Solution

We have the following lengths:

- Radius, $latex r=6$
- Height, $latex h=8$

Using the formula for the volume with these lengths, we have:

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

$latex V=\pi {{(6)}^2}\times 8$

$latex V=\pi (36)\times 8$

$latex V=904.8$

The volume is equal to 904.8 in³.

**EXAMPLE **5

**EXAMPLE**

What is the surface area of a cylinder that has a height of 12 yards and a radius of 8 yards?

##### Solution

We have the following lengths:

- Radius, $latex r=8$
- Height, $latex h=12$

Using the formula for the surface area, we have:

$latex A_{s}=2\pi r(r+h)$

$latex A_{s}=2\pi (8)(8+12)$

$latex A_{s}=2\pi (8)(20)$

$latex A_{s}=1005.3$

The surface area is equal to 1005.3 yd².

**EXAMPLE **6

**EXAMPLE**

Find the volume of a cube that has a diameter of 8 inches and a height of 12 inches.

##### Solution

We have the following lengths:

- Diameter, $latex d=8$
- Height, $latex h=12$

In this case, we have the length of the diameter, so we use the formula for volume in terms of the diameter of the cylinder:

$latex V=\pi (\frac{{{d}^2}}{4})\times h$

$latex V=\pi (\frac{{{(8)}^2}}{4})\times 12$

$latex V=\pi (\frac{64}{4})\times 12$

$latex V=\pi (16)\times 12$

$latex V=603.2$

The volume is equal to 603.2 in³.

**EXAMPLE **7

**EXAMPLE**

What is the surface area of a cylinder that has a diameter of 6 feet and a height of 7 feet?

##### Solution

Since we have the length of the diameter, we can simply divide the diameter by 2 to get the radius of the cylinder. Therefore, we have:

- Radius, $latex r=3$
- Height, $latex h=7$

Applying the surface area formula, we have:

$latex A_{s}=2\pi r(r+h)$

$latex A_{s}=2\pi (3)(3+7)$

$latex A_{s}=2\pi (3)(10)$

$latex A_{s}=188.5$

The surface area is equal to 188.5 ft².

**EXAMPLE **8

**EXAMPLE**

Find the volume of a cylinder that has a diameter of 12 feet and a height of 11 feet.

##### Solution

We have the following:

- Diameter, $latex d=12$
- Height, $latex h=11$

Using the formula for volume in terms of diameter, we have:

$latex V=\pi (\frac{{{d}^2}}{4})\times h$

$latex V=\pi (\frac{{{(12)}^2}}{4})\times 11$

$latex V=\pi (\frac{144}{4})\times 11$

$latex V=\pi (36)\times 11$

$latex V=1244.1$

The volume is equal to 1244.1 ft³.

**EXAMPLE **9

**EXAMPLE**

Find the surface area of a cylinder that has a diameter of 12 inches and a height of 13 inches.

##### Solution

We can divide the diameter by 2 to get the radius, and we have:

- Radius, $latex r=6$
- Height, $latex h=13$

Using the surface area formula with the given values, we have:

$latex A_{s}=2\pi r(r+h)$

$latex A_{s}=2\pi (6)(6+13)$

$latex A_{s}=2\pi (6)(19)$

$latex A_{s}=716.3$

The surface area is equal to 716.3 in².

**EXAMPLE **10

**EXAMPLE**

What is the volume of a hollow cylinder that has a height of 15 inches, an internal radius of 5 inches, and an external radius of 6 inches?

##### Solution

We have the following lengths:

- Height, $latex h=15$
- External radius, $latex r_{1}=6$
- Internal radius, $latex r_{2}=5$

Using the formula for the volume of a hollow cylinder, we have:

$latex V=\pi h({{r_{1}}^2}-{{r_{2}}^2})$

$latex V=\pi (15)({{6}^2}-{{5}^2})$

$latex V=\pi (15)(36-25)$

$latex V=\pi (15)(11)$

$latex V=518.4$

The volume is equal to 518.4 in³.

## Area and volume of a cylinder – Practice problems

Solve the following problems using the different formulas for the area and volume of a cylinder. You can use the solved examples above as a guide.

## See also

Interested in learning more about the area and volume of geometric figures? Take a look at these pages:

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