## Volume of a cylinder — Math Open Reference

Volume enclosed by a cylinder

Definition:

The number of cubic units that will exactly fill a cylinder

Try this

Drag the orange dot to resize the cylinder. The volume is calculated as you drag.

### How to find the volume of a cylinder

Although a cylinder is technically not a prism, it shares many of the properties of a prism. Like prisms,

the volume is found by multiplying the area of one end of the cylinder (base) by its height.

Since the end (base) of a cylinder is a circle, the area of that circle is given by the formula:

Multiplying by the height *h* we get

where:*π* is Pi, approximately 3.142*r* is the radius of the circular end of the cylinder*h* height of the cylinder

### Calculator

Use the calculator on the right to calculate height, radius or volume of a cylinder.

Enter any two values and the missing one will be calculated.

For example: enter the radius and height, and press ‘Calculate’. The volume will be calculated.

Similarly, if you enter the height and volume, the radius needed to get that volume will be calculated.

### Volume of a partially filled cylinder

One practical application is where you have horizontal cylindrical tank partly filled with liquid. Using the formula above you can find the volume of the cylinder which gives it’s maximum capacity, but you often need to know the volume of liquid in the tank given the depth of the liquid.

This can be done using the methods described in

Volume of a horizontal cylindrical segment.

### Oblique cylinders

Recall that an

oblique cylinder

is one that ‘leans over’ — where the top center is not over the base center point.

In the figure above check «allow oblique’ and drag the top orange dot sideways to see an oblique cylinder.

It turns out that the volume formula works just the same for these. You must however use the perpendicular height in the formula. This is the vertical line to left in the figure above.

To illustrate this, check ‘Freeze height’. As you drag the top of the cylinder left and right, watch the volume calculation and note that the volume never changes.

See Oblique Cylinders

for a deeper discussion on why this is so.

### Units

Remember that the radius and the height must be in the same units — convert them if necessary. The resulting volume will be in those cubic units.

So, for example if the height and radius are both in centimeters, then the volume will be in cubic centimeters.

### Things to try

- In the figure above, click ‘reset’ and ‘hide details’
- Drag the two dots to alter the size and shape of the cylinder
- Calculate the volume of that cylinder
- Click ‘show details’ to check your answer.

**While you are here..**

… I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone.

However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site?

*When we reach the goal I will remove all advertising from the site.*

It only takes a minute and any amount would be greatly appreciated.

Thank you for considering it! * – John Page*

Become a patron of the site at patreon.com/mathopenref

### Related topics

(C) 2011 Copyright Math Open Reference. All rights reserved

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## Volume of a cylinder segment

Definition: A shape formed when a cylinder is cut by a plane parallel to the sides of the cylinder.

Try this

Drag the orange dots, note how the volume changes.

If we take a horizontal cylinder, and cut it into two pieces using a cut parallel to the sides of the cylinder, we get two horizontal cylinder segments.

In the figure above, the bottom one is shown colored blue. The other one is the transparent part on top.

If we look a the end of the cylinder, we see it is a circle cut into two circle segments.

See Circle segment definition for more.

Whenever we have a solid whose cross-section is the same along its length, we can always find its volume by multiplying the area of the end by its length. So in this case, the volume of the cylinder segment is the area of the circle segment, times the length.

So as a formula the volume of a horizontal cylindrical segment is

Where

*s* = the area of the circle segment forming the end of the solid, and

*l* = the length of the cylinder.

The area of the circle segment can be found using it’s height and the radius of the circle.

See Area of a circle segment given height and radius.

### Calculator

Use the calculator below to calculate the volume of a horizontal cylinder segment.

It has been set up for the practical case where you are trying to find the volume of liquid is a cylindrical tank

by measuring the depth of the liquid.

For convenience, it converts the volume into liquid measures like gallons and liters if you select the desired units.

If you do not specify units the volume will be in whatever units you used to input the dimensions. For example, if you used

feet, then the volume will be in cubic feet. Use the same units for all three inputs.

### As a formula

where:*R* is the **radius** of the cylinder.*D* is the depth.*L* is the length of the cylinder

**Notes**:

- The result of the cos
^{-1}function in the formula is in radians. - The formula uses the radius of the cylinder. This is half its diameter.
- All inputs must be in the same units. The result will be in those cubic units.

So for example if the inputs are in inches, the result will be in cubic inches.

If necessary the result must be converted to liquid volume units such as gallons.

**While you are here..**

… I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone.

However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site?

*When we reach the goal I will remove all advertising from the site.*

It only takes a minute and any amount would be greatly appreciated.

Thank you for considering it! * – John Page*

Become a patron of the site at patreon.com/mathopenref

### Related topics

(C) 2011 Copyright Math Open Reference. All rights reserved

www.mathopenref.com

## Volume of a Cylinder

Calculating the volume of a cylinder is very similar to determining the area of a prism. You can think of a cylinder as being made up of many disks.

So if we find the area of one disk and multiply by the number of disks, we have the volume.

Recall that the area of a circle is equal to pi times the radius squared. And the number of disks can be thought of as the height. This gives us a volume formula.

Let’s give the formula a try.

V = Πr^{2}h

V = Π(3.6)^{2}(4) Here we started by replaced r with the radius and h with the height.

V = Π(12.96)(4) Using the order of operations, the radius was squared first.

V = 51.84Π Then we multiplied by 4. This is the answer in terms of pi.

V ≈ 162.86 in.^{3} After multiplying by Π, the answer was rounded to the nearest hundredth.

Here is another example:

Notice that in this example, the diameter is given instead of the radius.

Divide the diameter in half to get the radius. 30 mm ÷ 2 = 15 mm

Next, plug the values into the formula and solve.

V = Πr^{2}h

V = Π(15)^{2}(24)

V = Π(225)(24)

V = 5400Π mm^{3} ← If you are looking for an exact answer, STOP here!

V ≈ 16,964.6 mm^{3} ← Otherwise, you can round your answer to get an approximate. This solution is rounded to the nearest tenth.

Don’t be tricked by cylinders on their sides!

If it helps, try drawing a new sketch of the cylinder with the circular base on the bottom.

Now it may be clearer to see that the diameter of the base is 6 m and the height of the cylinder is 18 m. Use these measurements to calculate the volume. Be sure to first cut the diameter in half to determine the radius.

6m ÷ 2 = 3m Therefore, the radius is 3 meters.

V = Πr^{2}h

V = Π(3)^{2}(18)

V = Π(9)(18)

V = 162Π m^{3} ← Remember that this is the exact answer.

V ≈ 509 m^{3} ← This is an approximate answer, rounded to the

nearest whole.

**Let’s Review**

To calculate the volume of a cylinder, multiply the area of the base (Πr^{2}) times the height (h). This gives the formula V = Πr^{2}h. Because pi is an irrational number, you may need to round your answer. If an exact answer is needed, leave the pi in the solution. Other key things to notice when solving include determining whether you have the diameter or the radius and if the cylinder is upright or not. If you have the diameter, divide the length in half to determine the radius. If the cylinder is on its side, you could redraw the object the make sure that you have correctly labeled which length is the radius and which is the height. Then you are ready to solve!

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## Finding the Volume and Surface Area of a Cylinder

#### Learning Objectives

- Find the volume and surface area of a cylinder

If you have ever seen a can of soda, you know what a cylinder looks like. A cylinder is a solid figure with two parallel circles of the same size at the top and bottom. The top and bottom of a cylinder are called the bases. The height [latex]h[/latex] of a cylinder is the distance between the two bases. For all the cylinders we will work with here, the sides and the height, [latex]h[/latex] , will be perpendicular to the bases.

A cylinder has two circular bases of equal size. The height is the distance between the bases.

Rectangular solids and cylinders are somewhat similar because they both have two bases and a height. The formula for the volume of a rectangular solid, [latex]V=Bh[/latex] , can also be used to find the volume of a cylinder.

For the rectangular solid, the area of the base, [latex]B[/latex] , is the area of the rectangular base, length × width. For a cylinder, the area of the base, [latex]B[/latex], is the area of its circular base, [latex]\pi {r}^{2}[/latex]. The image below compares how the formula [latex]V=Bh[/latex] is used for rectangular solids and cylinders.

Seeing how a cylinder is similar to a rectangular solid may make it easier to understand the formula for the volume of a cylinder.

To understand the formula for the surface area of a cylinder, think of a can of vegetables. It has three surfaces: the top, the bottom, and the piece that forms the sides of the can. If you carefully cut the label off the side of the can and unroll it, you will see that it is a rectangle. See the image below.

By cutting and unrolling the label of a can of vegetables, we can see that the surface of a cylinder is a rectangle. The length of the rectangle is the circumference of the cylinder’s base, and the width is the height of the cylinder.

The distance around the edge of the can is the circumference of the cylinder’s base it is also the length [latex]L[/latex] of the rectangular label. The height of the cylinder is the width [latex]W[/latex] of the rectangular label. So the area of the label can be represented as

To find the total surface area of the cylinder, we add the areas of the two circles to the area of the rectangle.

The surface area of a cylinder with radius [latex]r[/latex] and height [latex]h[/latex], is

[latex]S=2\pi {r}^{2}+2\pi rh[/latex]

#### Volume and Surface Area of a Cylinder

For a cylinder with radius [latex]r[/latex] and height [latex]h:[/latex]

#### example

A cylinder has height [latex]5[/latex] centimeters and radius [latex]3[/latex] centimeters. Find the 1. volume and 2. surface area.

Solution

Step 1. Read the problem. Draw the figure and labelit with the given information. |

1. | |

Step 2. Identify what you are looking for. | the volume of the cylinder |

Step 3. Name. Choose a variable to represent it. | let V = volume |

Step 4. Translate.Write the appropriate formula. Substitute. (Use [latex]3.14[/latex] for [latex]\pi [/latex] ) | [latex]V=\pi {r}^{2}h[/latex] [latex]V\approx \left(3.14\right){3}^{2}\cdot 5[/latex] |

Step 5. Solve. | [latex]V\approx 141.3[/latex] |

Step 6. Check: We leave it to you to check your calculations. | |

Step 7. Answer the question. | The volume is approximately [latex]141.3[/latex] cubic inches. |

2. | |

Step 2. Identify what you are looking for. | the surface area of the cylinder |

Step 3. Name. Choose a variable to represent it. | let S = surface area |

Step 4. Translate.Write the appropriate formula. Substitute. (Use [latex]3.14[/latex] for [latex]\pi [/latex] ) | [latex]S=2\pi {r}^{2}+2\pi rh[/latex] [latex]S\approx 2\left(3.14\right){3}^{2}+2\left(3.14\right)\left(3\right)5[/latex] |

Step 5. Solve. | [latex]S\approx 150.72[/latex] |

Step 6. Check: We leave it to you to check your calculations. | |

Step 7. Answer the question. | The surface area is approximately [latex]150.72[/latex] square inches. |

#### example

Find the 1. volume and 2. surface area of a can of soda. The radius of the base is [latex]4[/latex] centimeters and the height is [latex]13[/latex] centimeters. Assume the can is shaped exactly like a cylinder.

Show Answer

Solution

Step 1. Read the problem. Draw the figure andlabel it with the given information. |

1. | |

Step 2. Identify what you are looking for. | the volume of the cylinder |

Step 3. Name. Choose a variable to represent it. | let V = volume |

Step 4. Translate.Write the appropriate formula. Substitute. (Use [latex]3.14[/latex] for [latex]\pi [/latex] ) | [latex]V=\pi {r}^{2}h[/latex] [latex]V\approx \left(3.14\right){4}^{2}\cdot 13[/latex] |

Step 5. Solve. | [latex]V\approx 653.12[/latex] |

Step 6. Check: We leave it to you to check. | |

Step 7. Answer the question. | The volume is approximately [latex]653.12[/latex] cubic centimeters. |

2. | |

Step 2. Identify what you are looking for. | the surface area of the cylinder |

Step 3. Name. Choose a variable to represent it. | let S = surface area |

Step 4. Translate.Write the appropriate formula. Substitute. (Use [latex]3.14[/latex] for [latex]\pi [/latex] ) | [latex]S=2\pi {r}^{2}+2\pi rh[/latex] [latex]S\approx 2\left(3.14\right){4}^{2}+2\left(3.14\right)\left(4\right)13[/latex] |

Step 5. Solve. | [latex]S\approx 427.04[/latex] |

Step 6. Check: We leave it to you to check your calculations. | |

Step 7. Answer the question. | The surface area is approximately [latex]427.04[/latex] square centimeters. |

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## How to Find the Volume of a Cylinder

### Cylinder – Definition

Cylinder is one of the basic conic shapes found in geometry, and its properties have been known for thousands of years. In general a cylinder is defined as the set of points that lies at a constant distance from a line segment, where the line segment is known as the axis of the cylinder.

In a broader sense, a cylinder can be defined as a curved surface formed by a line segment parallel to another line segment, when travelling in a path defined by some geometrical equation. This definition allows the inclusion of several other types of cylinders into to create a cylinder family. If the cross section is an ellipse, the cylinder is an elliptical cylinder. If the cross section is a parabola or a hyperbola,it is referred to as parabolic and hyperbolic cylinders respectively.

A circular cylinder can be considered as a limiting case of the n sided prisms, where n reaches infinity.

In general, the fixed line described above serves as the axis of the cylinder and either of the planar surfaces is referred to as bases. The perpendicular distance between the bases is known as the height of the cylinder.

### Using the Formula to Find the Volume of a Cylinder

For a general cylinder with a base area A and height h, the volume of the cylinder is given by the formula:

**V _{cylinder}=Ah **

If the cylinder has circular cross section, the equation reduces to

**V=πr ^{2} h**

where r is the radius. Even if the shapes of the cylinders are not regular, i.e. cylinders bases are not forming right angles with the curved surface, the above equations hold.

To find the volume of a cylinder, one should know two things,

**Height of the cylinder****The area of the cross-section –**If the cylinder has circular cross section, the radius has to be known. To determine the area of elliptical or parabolic or hyperbolic, other information is needed to determine the area, and further calculation has to be carried out.

**Calculating the Volume of a Cylinder – Examples**

- The inner radius of a cylindrical water tank is 3m. If the water is filled to a height of 1.5m find the volume of water included in the tank.

The radius of the base is given as 3m and the height as 1.5m. Therefore, applying the volume of a cylinder formula, we can get the volume of water in the tank.

**V=πr ^{2} h=3.14×3^{2}×1.5=42.39m^{3}**

- A cylindrical fuel tank has a diameter of 6m and a length of 20m fuel, the tank is filled only 80% of its capacity. If a motor empties the tank in 1 hour and 40 minutes, find the average volume transfer rate of the pump.

To find the volume transfer rate of the pump, the total volume pumped out has to be determined. Therefore, it is needed to calculate the volume of the tank. Since the diameter is given, we can determine the radius by formula D=2r. The radius is 3m. Using the volume of a cylinder formula we have

**V=πr ^{2} h=3.14×3^{2}×20=565.2m^{3}**

Volume of the fuel inside is only 80 of the total volume and it has taken 100 minutes to empty the tank, the volume flow rate is

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## Volume of a Cylinder (Worksheets)

The worksheets below can be used to help teach students how to calculate the volumes of cylinders. The *basic* level worksheets do not include decimal measurements. The *advanced* level includes decimals and/or fractional measurements.

These worksheets align with Math Common Core Standard 8.G.9.

Calculate the volume of each cylinder. There is an example and formula at the top of the page.

7th and 8th Grades

This worksheet shows 9 cylinders. The height and radius of the bases are given. Students calculate the volume of each.

7th and 8th Grades

Compare the volumes of two cylinders. Answer 14 questions that walk you through volume calculations step-by-step.

7th and 8th Grades

Calculate the volumes of the cylinders in each of the word problems.

7th and 8th Grades

This file has 30 task cards that you can cut apart and use for small group instruction, peer study groups, learning centers, classroom scavenger hunts, and classroom games.

7th and 8th Grades

The top of this page is an explanation of how to find the volume of a cylinder. Beneath that are six problems for students to solve.

7th and 8th Grades

Calculate the volume of 9 different cylinders. In this set, one or more of the measurements given is a decimal.

8th Grade

Solve each word problem by finding volumes of cylinders. Problems include decimals up to hundredths.

7th and 8th Grades

Volumes of Rectangular Prisms

From this page you can download a collection of worksheets on calculating the volumes of rectangular prisms.

www.superteacherworksheets.com

## Volume of a hollow cylinder Calculator

- Purpose of use
- Calculate the solid volume of a hollow tube of quartz glass so that I can estimate its mass (knowing density).
- Comment/Request
- Thank you.

[1] 2018/12/06 01:49 Male / 40 years old level / An engineer / Very /

- Purpose of use
- Calculate the volumes of cement needed to construct a foam cored fake piling for aquarium display.

[2] 2018/11/21 08:21 Female / 30 years old level / Others / Very /

- Purpose of use
- Math homwurk

[3] 2018/11/13 11:05 Male / 60 years old level or over / Elementary school/ Junior high-school student / A little /

- Purpose of use
- Calculate the needed volume of solder plated in a circuit board plated through hole that will result in 50% min fill after the reflow operation. Made an estimated of what the solder meniscus will look like after reflow.

[4] 2018/10/24 06:16 Male / 50 years old level / An engineer / Very /

- Purpose of use
- Calculate the surface area of a segment of a urinary catheter
- Comment/Request
- The formula for calculating the final surface area is wrong. It should be S = F + pi(r1^2 — r2^2) NOT S = F + 2pi(r1^2 — r2^2) becasue the surface area of a circle is pi*r^2, not 2pi*r^2
- from Keisan
- The hollow cylinder has a top surface and a bottom surface. (pi*r^2 * 2)

[5] 2018/10/16 03:24 Male / 20 years old level / High-school/ University/ Grad student / Not at All /

- Purpose of use
- Partial volume of a roll of paper

[6] 2018/09/21 22:50 Male / 30 years old level / An engineer / Very /

- Purpose of use
- To save me typing volume formula into calculator (because I»m lazy!)

[7] 2018/09/03 21:42 Male / 20 years old level / An engineer / Very /

- Purpose of use
- Engineering Calculations

[8] 2018/08/03 03:53 Male / Under 20 years old / High-school/ University/ Grad student / Very /

- Purpose of use
- Finding the area of fuel grain needed for my hybrid rocket motor in development

[9] 2018/06/27 12:58 Male / 20 years old level / High-school/ University/ Grad student / Very /

- Purpose of use
- Fantasy gaming calculation
- Comment/Request
- What are you looking at? Old people do roleplay too. And get off my lawn.

[10] 2018/06/13 07:44 Male / 50 years old level / An engineer / Very /

keisan.casio.com

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