# Volume Of A Cylinder Formula

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## Volume of a Cylinder

The volume of a cylinder is the density of the cylinder which signifies the amount of material it can carry or how much amount of any material can be immersed in it. Cylinder’s volume is given by the formula, πr2h, where r is the radius of the circular base and h is the height of the cylinder. The material could be a liquid quantity or any substance which can be filled in the cylinder uniformly.

Volume of cylinder has been explained in this article briefly along with solved examples for better understanding. In Mathematics, geometry is an important branch where we learn the shapes and their properties. Volume and surface area are the two important properties of any 3d shape.

A cylinder is a solid composed of two congruent circles in parallel planes, their interiors, and all the line segments parallel to the segment containing the centers of both circles with endpoints on the circular regions.

The volume of a 3 -dimensional solid is the amount of space it occupies.  Volume is measured in cubic units ( in3,ft3,cm3,m3 , et cetera).  Be sure that all of the measurements are in the same unit before computing the volume.

The volume V of a cylinder with radius r is the area of the base B times the height h

.V=Bh  or  V=πr2h

## Definition

The cylinder is a three-dimensional shape having a circular base. A cylinder can be seen as a set of circular disks that are stacked on one another. Now, think of a scenario where we need to calculate the amount of sugar that can be accommodated in a cylindrical box.

In other words, we mean to calculate the capacity or volume of this box. The capacity of a cylindrical box is basically equal to the volume of the cylinder involved. Thus, the volume of a three-dimensional shape is equal to the amount of space occupied by that shape.

## Volume of a Cylinder Formula

A cylinder can be seen as a collection of multiple congruent disks, stacked one above the other. In order to calculate the space occupied by a cylinder, we calculate the space occupied by each disk and then add them up. Thus, the volume of the cylinder can be given by the product of the area of base and height.

For any cylinder with base radius ‘r’, and height ‘h’, the volume will be base times the height.

Therefore, the cylinder’s volume of base radius ‘r’, and height ‘h’ = (area of base) × height of the cylinder

Since the  base is the circle, it can be written as

Volume =  πr× h

Therefore, the volume of a cylinder = πr2h cubic units.

### Volume of Hollow Cylinder

In case of hollow cylinder, we measure two radius, one for inner circle and one for outer circle formed by the base of hollow cylinder. Suppose, r1 and r2 are the two radii of the given hollow cylinder with ‘h’ as the height, then the volume of this cylinder can be written as;

• V =  πh(r12 – r22)

### Surface Area of Cylinder

The amount of square units required to cover the surface of the cylinder is the surface area of the cylinder. The formula for the surface area of the cylinder is equal to the total surface area of the bases of the cylinder and surface area of its sides.

• A = 2πr2 + 2πrh

#### Volume of Cylinder in Litres

When we find the volume of the cylinder in cubic centimetres, we can convert the value in litres by knowing the below conversion, i.e.,

1 Litre = 1000 cubic cm or cm3
For example: If a cylindrical tube has a volume of 12 litres, then we can write the volume of the tube as 12 × 1000 cm3 = 12,000 cm3

### Examples

Question 1: Calculate the volume of a given cylinder having height 20 cm and base radius of 14 cm. (Take pi = 22/7)

Solution:

Given:

Height  = 20 cm

we know that;

Volume, V = πr2h  cubic units

V=(22/7) × 14  × 14  × 20

V= 12320 cm3

Therefore, the volume of a cylinder = 12320 cm3

Question 2: Calculate the radius of the base of a cylindrical container of volume 440 cm3. Height of the cylindrical container is 35 cm. (Take pi = 22/7)

Solution:

Given:

Volume = 440 cm3

Height = 35 cm

We know from the formula of cylinder;

Volume, V = πr2h  cubic units

So, 440 = (22/7) × r2 × 35

r= (440 × 7)/(22 × 35) = 3080/770 = 4

Therefore, r = 2 cm

Therefore, the radius of a cylinder = 2 cm.

## Frequently Asked Questions on Volume of a Cylinder

### What is meant by the volume of a cylinder?

In geometry, the volume of a cylinder is defined as the capacity of the cylinder, which helps to find the amount of material that the cylinder can hold.

### What is the formula for the volume of a cylinder?

The formula to calculate volume of a cylinder is given by the product of base area and its height.
Since, the base area of a cylinder is circular, we can state that
Volume of a cylinder = πr2h cubic units.

### What is the volume of a hollow cylinder?

As we know, the hollow cylinder is a type of cylinder, which is empty from inside and it should possess some difference between the internal and the external radius. Thus, the amount of space occupied by the hollow cylinder in the three dimensional space is called the volume of a hollow cylinder.

### How to calculate the volume of a hollow cylinder?

If R is the external radius and r is the internal radius, then the formula for calculating the cylinder’s volume is given by:
V = π (R2 – r2) h cubic units.

### What is the unit for the volume of a cylinder?

The volume of a cylinder is generally measured in cubic units, such as cubic centimeters (cm3), cubic meters (m3), cubic feet (ft3) and so on.

### How to find the volume of a cylinder if the diameter and height are given?

As we know, Diameter “d” = 2(Radius) = 2r.
So, r = d/2
Now, substitute the value of “r” in the volume of cylinder formula, we get
V = πr2h = π(d/2)2h
V = (πd2h)/4
Hence, the volume of the cylinder is (πd2h)/4, if its diameter and height are given.

### What will happen to the cylinder’s volume if its radius is doubled?

As we know, cylinder’s volume is directly proportional to the square of its radius.
If the radius is doubled, (i.e., r = 2r), we get
V = πr2h =π(2r)2h = 4πr2h.
Hence, the cylinder’s volume becomes four times, when its radius is doubled.

### What will happen to the cylinder’s volume if its radius is halved?

We know that, the volume of cylinder ∝ Radius2
Thus, if radius is halved, (i.e., r = r/2), we get
V = π(r/2)2h = (πr2h)/4
Therefore, the cylinder’s volume becomes 1/4th, if its radius is halved.

Example:

Find the volume of the cylinder shown. Round to the neatest cubic centimeter.

Solution

The formula for the volume of a cylinder is V=Bh  or  V=πr2h

.The radius of the cylinder is 8 cm and the height is 15cm.

Substitute 8 for r and 15 for h in the formula V=πr2h

.V=π(8)2(15)

Simplify.

V=π(64)(15)≈3016

Therefore, the volume of the cylinder is about 3016 cubic centimeters.

## How to calculate volume of a cylinder?

Let’s start from the beginning – what is a cylinder? It’s a solid bounded by a cylindrical surface and two parallel planes. We can imagine it as a solid physical tin having lids on top and bottom. To calculate its volume, we need to know two parameters – the radius (or diameter) and height:

`cylinder volume = π × cylinder radius² × cylinder height`

The cylinder volume calculator helps in finding the volume of right, hollow and oblique cylinders:

## Volume of a hollow cylinder

The hollow cylinder, also called the cylindrical shell, is a three-dimensional region bounded by two right circular cylinders having the same axis and two parallel annular bases perpendicular to the cylinders’ common axis.

It’s easier to understand that definition by imagining, e.g., a drinking straw or a pipe – the hollow cylinder is this plastic, metal, or other material. The formula behind the volume of a hollow cylinder is:

`cylinder_volume = π × (R² - r²) × cylinder_height`

where `R` – external radius, and `r` – internal radius

Similarly, we can calculate the cylinder volume using the external diameter, `D`, and internal diameter, `d`, of a hollow cylinder with this formula:

`cylinder_volume = π × [(D² - d²)/4] × cylinder_height`

To calculate the volume of a cylindrical shell, let’s take some real-life examples, maybe… a roll of toilet paper, because why not?

1. Enter the external diameter of the cylinder. The standard is equal to approximately 11 cm.
2. Determine the internal cylinder diameter. It’s the internal diameter of the cardboard part, around 4 cm.
3. Find out what’s the height of the cylinder; for us, it’s 9 cm.
4. Tadaaam! The volume of a hollow cylinder is equal to 742.2 cm³.

Remember that the result is the volume of the paper and the cardboard. If you want to calculate how much plasticine you can put inside the cardboard roll, use the standard formula for the volume of a cylinder – the calculator will calculate it in the blink of an eye!

## FAQ

### Where can you find cylinders in nature?

Cylinders are all around us, and we are not just talking about Pringles’ cans. Although things in nature are rarely perfect cylinders, some examples are tree trunks & plant stems, some bones (and therefore bodies), and the flagella of microscopic organisms. These make up a large amount of the natural objects on Earth!

### How do you draw a cylinder?

To draw a cylinder, follow these steps:

1. Draw a slightly flattened circle. The more flattened it is, the closer you are to looking at the cylinder side on.
2. Draw two equal, parallel lines from the far sides of your circle going down.
3. Link the ends of the two lines with a semi-circular line that looks the same as the bottom half of your top circle.

### How do you calculate the weight of a cylinder?

To calculate the weight of a cylinder:

1. Square the radius of the cylinder.
2. Multiply the square of radius by pi and the cylinder’s height.
3. Multiply the volume by the density of the cylinder. The result is the cylinder’s weight.

### How do you calculate the surface area to volume ratio of a cylinder?

1. Find the volume of the cylinder using the formula πr²h.
2. Find the surface area of the cylinder using the formula 2πrh + 2πr².
3. Make a ratio out of the two formulas, i.e., πr²h : 2πrh + 2πr².
4. Alternatively, simplify it to rh : 2(h+r).
5. Divide both sides by one of the sides to get the ratio in its simplest form.

### How do you find the height of a cylinder?

If you have the volume and radius of the cylinder:

1. Make sure the volume and radius are in the same units (e.g., cm³ and cm), and the radius is in radians.
3. Divide the volume by the radius squared and pi to get the height in the same units as the radius.

If you have the surface area and radius (r):

1. Make sure the surface and radius are in the same units and the radius is in radians.
2. Subtract 2πr² from the surface area.
3. Divide the result of step 1 by 2πr.
4. The result is the height of the cylinder.

### How do I find the radius of a cylinder?

If you have the volume and height of the cylinder:

1. Make sure the volume and height are in the same units (e.g., cm³ and cm), and the radius is in radians.
2. Divide the volume by pi and the height.
3. Square root the result.

If you have the surface area and height (h):

1. Substitute the height, h, and surface area into the equation, surface area = πr²h : 2πrh + 2πr².
2. Divide both sides by 2π.
3. Subtract surface area/2π from both sides.
4. Solve the resulting quadratic equation.
5. The positive root is the radius.

### How do you find the volume of a right trapezoidal cylinder?

A right trapezoidal cylinder, also known as a rectangular prism, can be solved as such:

1. Add the two parallel sides (bases) of the trapezium together.
2. Divide the result by 2.
3. Multiply the result of step 2 by the height of the trapezium (i.e., the distance that separates the two sides).
4. Multiply the result by the length of the cylinder.
5. The result is the area of a right trapezoidal cylinder.

### How do you find the volume of an oval cylinder?

To find the volume of an oval cylinder:

1. Multiply the smallest radius of the oval (minor axis) by its largest radius (major axis).
2. Multiply this new number by pi.
3. Divide the result of step 2 by 4. The result is the area of the oval.
4. Multiply the area of the oval by the height of the cylinder.
5. The result is the volume of an oval cylinder.

### How do you find the volume of a slanted cylinder?

To calculator the volume of a slanted cylinder:

1. Find the radius, side length, and slant angle of the cylinder.
3. Multiply the result by pi.
4. Take the sin of the angle.
5. Multiply the sin by the side length.
6. Multiply the result from steps 3 and 5 together.
7. The result is the slanted volume.

### How do you calculate the swept volVolume of Cylinderume of a cylinder?

To compute the swept volume of a cylinder:

1. Divide the bore diameter by 2 to get the bore radius.
3. Multiply the square radius by pi.
4. Multiply the result of step 3 by the length of the stroke. Make sure the units for bore and stroke length are the same.
5. The result is the swept volume of one cylinder.

## Volume of Cylinder

The volume of a cylinder is the capacity of the cylinder which calculates the amount of material quantity it can hold. In geometry, there is a specific volume of a cylinder formula that is used to measure how much amount of any quantity whether liquid or solid can be immersed in it uniformly. A cylinder is a three-dimensional shape with two congruent and parallel identical bases. There are different types of cylinders. They are:

• Right circular cylinder: A cylinder whose bases are circles and each line segment that is a part of the lateral curved surface is perpendicular to the bases.
• Oblique Cylinder: A cylinder whose sides lean over the base at an angle that is not equal to a right angle.
• Elliptic Cylinder: A cylinder whose bases are ellipses.
• Right circular hollow cylinder: A cylinder that consists of two right circular cylinders bounded one inside the other.

## What is the Volume of a Cylinder?

The volume of a cylinder is the number of unit cubes (cubes of unit length) that can be fit into it. It is the space occupied by the cylinder as the volume of any three-dimensional shape is the space occupied by it. The volume of a cylinder is measured in cubic units such as cm3, m3, in3, etc. Let us see the formula used to calculate the volume of a cylinder.

### Definition of a Cylinder

A cylinder is a three-dimensional solid shape that consists of two parallel bases linked by a curved surface. These bases are like a circular disk in a shape. The line passing from the center or joining the centers of two circular bases is called the axis of the cylinder.

## Volume of Cylinder Formula

We know that a cylinder resembles a prism (but note that a cylinder is not a prism as it has a curved side face), we use the same formula of volume of a prism to calculate the volume of a cylinder as well. We know that the area of a prism is calculated using the formula,

V = A × h, where

• A = area of the base
• h = height

Now we will apply this formula to calculate the volume of different types of cylinders.

## Volume of a Right Circular Cylinder

We know that the base of a right circular cylinder is a circle and the area of a circle of radius ‘r’ is πr2. Thus, the volume (V) of a right circular cylinder, using the above formula, is,

V = πr2h

• ‘r’ is the radius of the base (circle) of the cylinder
• ‘h’ is the height of the cylinder
• π is a constant whose value is either 22/7 (or) 3.142.

Here,

Thus, the volume of cylinder directly varies with its height and directly varies with the square of its radius. i.e., if the radius of the cylinder becomes double, then its volume becomes four times.

## Volume of an Oblique Cylinder

The formula to calculate the volume of cylinder (oblique) is the same as that of a right circular cylinder. Thus, the volume (V) of an oblique cylinder whose base radius is ‘r’ and whose height is ‘h’ is,

V = πr2h

## Volume of an Elliptic Cylinder

We know that an ellipse has two radii. Also, we know that the area of an ellipse whose radii are ‘a’ and ‘b’ is πab. Thus, the volume of an elliptic cylinder is,

V = πabh

Here,

• ‘a’ and ‘b’ are the radii of the base (ellipse) of the cylinder.
• ‘h’ is the height of the cylinder.
• π is a constant whose value is either 22/7 (or) 3.142.

## Volume of a Right Circular Hollow Cylinder

As a right circular cylinder is a cylinder that consists of two right circular cylinders bounded one inside the other, its volume is obtained by subtracting the volume of the inside cylinder from that of the outside cylinder. Thus, the volume (V) of a right circular hollow cylinder is,

V = π(R2 – r2)h

Here,

• ‘R’ is the base radius of the outside cylinder.
• ‘r’ is the base radius of the inside cylinder.
• ‘h’ is the height of the cylinder.
• π is a constant whose value is either 22/7 (or) 3.142.

## How To Calculate the Volume of Cylinder?

Here are the steps to calculate the volume of cylinder:

• Identify the radius to be ‘r’ and height to be ‘h’ and make sure that they both are of the same units.
• Substitute the values in the volume formula V = πr2h.
• Write the units as cubic units.

Example: Find the volume of a right circular cylinder of radius 50 cm and height 1 meter. Use π = 3.142.

Solution:

The radius of the cylinder is, r = 50 cm.

Its height is, h = 1 meter = 100 cm.

Its volume is, V = πr2h = (3.142)(50)2(100) = 785,500 cm3.

Note: We need to use the formula to find the volume of a cylinder depending on its type as we discussed in the previous section. Also, assume that a cylinder is a right circular cylinder if there is no type given and apply the volume formula to be V = πr2h.

## Volume of Cylinder Examples

Example 2: Find the volume of an elliptic cylinder whose base radii are 7 inches and 10 inches, and whose height is 15 inches. Use π = 22/7.

Solution:

The base radii of the given elliptic cylinder are,

a = 7 inches and b = 10 inches.

Its height is, h = 15 inches.

Using the volume of cylinder formula, the volume of the given elliptic cylinder is,

V = πabh

V = (22/7) × 7 × 10 × 15 = 3300 cubic inches.

Answer: The volume of the given cylinder is 3,300 cubic inches.

Example 3: What is the volume of the cylinder with a radius of 4 units and a height of 6 units?

Solution:

Radius,r = 4 units Height,h = 6 units

Volume of the cylinder, V = πr2h cubic units.

V = (22/7) × (4)2 × 6 V = 22/7 × 16 × 6

V= 301.71 Cubic units.

Therefore the volume of the cylinder is 301.71 cubic units.

## FAQs on Volume of Cylinder

### What is the Volume of Cylinder?

The volume of cylinder is the amount of space in it. It can be obtained by multiplying its base area by its height. The volume of a cylinder of base radius ‘r’ and height ‘h’ is V = πr2h.

### What Is the Formula for Calculating the Volume of a Cylinder?

The formula for calculating the volume of a cylinder is V = πr2h, where

• ‘r’ is the radius of the base of the cylinder
• ‘h’ is the height of the cylinder
• π is a constant whose value is either 22/7 (or) 3.142.

### What is the Volume of a Cylinder with Diameter?

Let us consider a cylinder of radius ‘r’, diameter ‘d’, and height ‘h’. The volume of a cylinder of base radius ‘r’ and height ‘h’ is V = πr2h. We know that r = d/2. By substituting this in the above formula, V = πd2h/4.

### What Is the Ratio of the Volume of a Cylinder and a Cone?

Let us consider a cylinder and a cone, each with base radius ‘r’ and height ‘h’. We know that the volume of the cylinder is πr2h and the volume of the cone is 1/3 πr2h. Thus the required ratio is 1:(1/3) (or) 3:1.

### How To Find Volume of a Cylinder With Diameter and Height?

The volume of a cylinder with base radius ‘r’ and height ‘h’ is, V = πr2h. If its base diameter is d, then we have d = r/2. Substituting this in the above formula, we get V = πd2h/4. Thus, the formula to find the volume of a cylinder with the diameter (d) and height(h) is V = πd2h/4.

### How To Find Volume of Cylinder With Circumference and Height?

We know that the circumference of a circle of radius ‘r’ is C = 2πr. Thus, when the circumference of the base of a cylinder (C) and its height (h) are given, then we first solve the equation C = 2πr for ‘r’ and then we apply the volume of a cylinder formula, which is, V = πr2h.

### How To Calculate Volume of Cylinder in Litres?

We can use the following conversion formulas to convert the volume of cylinder from m3 (or) cm3 to liters.

• 1 m3 = 1000 liters
• 1 cm3 = 1 ml (or) 0.001 liters

### What Happens to the Volume of Cylinder When Its Radius Is Halved?

The volume of cylinder varies directly with the square of its radius. Thus, when its radius is halved, the volume becomes 1/4th.

### What Happens to the Volume of Cylinder When Its Radius Is Doubled?

We know that the volume of cylinder is directly proportional to the square of its radius. Thus, when its radius is doubled, the volume becomes four times.

### How Do You Find Volume of Cylinder Using Calculator?

Volume of a cylinder calculator is a machine to calculate a cylinder’s volume. To calculate the volume of a cylinder using a calculator we need to provide necessary inputs to the calculator tool, such as required dimensions like radius, diameter, height, etc. Try now the volume of a cylinder calculator enter the radius and height of the cylinder in the given box of the volume of a cylinder calculator. Click on the “Calculate” button to find the volume of a cylinder. By clicking the “Reset” button you can easily clear the previously entered data and find the volume of a cylinder for different values.

### What is the Area and Volume of a Cylinder?

The surface area of a cylinder is the total area or region covered by the surface of the cylinder. The surface area of a cylinder is given by two following formulas:

• The curved surface area of cylinder = 2πrh
• The total surface area of the cylinder = 2πr2+2πrh = 2πr(h+r)

The area of a cylinder is expressed in square units, like m2, in2, cm2, yd2, etc.

The volume of a cylinder is the total amount of capacity immersed in a cylinder that can be calculated using the volume formula for the cylinder which is V = πr2h. The volume of a cylinder is always measured in cubic units.

### How Does the Volume of a Hollow Cylinder Change When the Height is Doubled?

The volume of a hollow cylinder formula is V = π(R2 – r2)h cubic units. According to the volume formula, we can see that volume is directly proportional to the height of the hollow cylinder. Therefore, the volume gets doubled when the height of the hollow cylinder is doubled.

### What is the Volume of Cylinder in Terms of Pi?

The volume of cylinder is defined as the capacity of a cylinder which is indicated in terms of pi. The volume of a cylinder in terms of pi is expressed in cubic units where units can be m3, cm3, in3, or ft3.

Examples to Find The Volume of a Cylinder

Example 1: A cylinder has a radius of 50 cm and a height of 100 cm. How to find the volume of a cylinder?

Solution: We know the volume of a cylinder is given by the formula – π r2 h, where r is the radius of the cylinder and h is the height.

Therefore, putting the values, we get,

V = π r2 h

= 3.14 x 502 x 100 = 785,000 cm3

Example 2: How do you find the volume of a cylinder whose one of the radii is 40 cm and another is 60 cm? The cylinder has a height of 200 cm.

Solution: From the data given, you can find that the cylinder is elliptical as the radii are different. To find the volume of an elliptical cylinder, the formula is V = π abh, where a, and b are radii, and h is the height.

Therefore, the volume of a cylinder = V = π abh

= π x 40 x 60 x 200 = 1507200 cm3

Example 3: How do you find the volume of a hollow cylinder from the inside and has outer and inner radii of units 6 and 8, respectively? The height of this hollow cylinder is 15 units.

Solution: We know the formula for the volume of a hollow cylinder is given by V = π (R2 – r2) h.

Therefore, putting the values, we get,

V = π (R2 – r2) h

= π (82 – 62) 15 = 1318.8 units2

Example 4: One day, Alex was wondering, “How do I find the volume of a cylinder whose height is 6 inches and radius is 3 inches.” Can you help her to find the volume of that cylinder?

Answer: Yes, you can! You know the formula to find the volume of a cylinder is given by: V = π r2 h.

Therefore, by putting the values, you get, V = π r2

= π x 32 x 6 = 169.56 in3

You can tell Alex that the volume of the cylinder is 169.56 in3

## Frequently Asked Questions – FAQs

1. What is the curved surface area of cylinder?

Ans) The curved surface area of cylinder = 2πrh

2. What is the Volume of Cylinder?

Ans) The volume of cylinder is the amount of space in it. It can be obtained by multiplying its base area by its height. The volume of a cylinder of base radius ‘r’ and height ‘h’ is V = πr2h.

3. What is the Volume of Hollow Cylinder?

Ans) We measure two radii for the volume of a hollow cylinder one for the inner circle and the other for the outer circle created by the hollow cylinder’s base and If “R” is the outer radius and “r” is the inner radius and “h” is the height, then the volume of the hollow cylinder is V = πh (R2 – r2).

4. What is the unit for the volume of a cylinder?

Ans) The volume of a cylinder is measured in cubic units, such as cubic centimeters (cm3), cubic meters (m3), cubic feet (ft3) and so on.

## Practice Question

1. Calculate the volume of a given cylinder having height 30 cm and base radius of 15 cm. (Take pi = 22/7)?

Ans) Given:

Height  = 30 cm

we know that;

Volume, V = πr2h  cubic units

V=(22/7) × 15  × 15  × 30

V= 212142.85 cm3

Therefore, the volume of a cylinder = 212142.85 cm3

2. A cylinder has a height of 15cm and a volume of 500cm3 ! What is the radius of the cylinder?

Ans) 3.257

### Example 1: Finding the Volume of a Cylinder given the Radius and Height

Find the volume of the given cylinder, rounded to the nearest tenth.

From the diagram, we see that the cylinder has a radius of 4.2 ft and a height of 6.5 ft. Recalling that the volume, 𝑉, of a cylinder of radius 𝑟 and height ℎ is given by the formula

We were asked to round our answer to the nearest tenth. Remember that the tenths digit is the first digit after the decimal point, which in this case is a 2. The digit following this (i.e., the hundredths digit) is a 1, so the answer rounds down to 360.2 to the nearest tenth.

Since the radius and height of the cylinder were given in feet, the volume must be in cubic feet. The volume of the cylinder, rounded to the nearest tenth, is 360.2 ft3.

We will also look at an example where we are given the diameter of the cylinder, not the radius. Our approach here is very similar but with one additional step. Always be careful to check if you are given a radius or a diameter in the question.

### Example 2: Finding the Volume of a Cylinder given the Diameter and Height

Find, to the nearest tenth, the volume of this cylinder.

First, recall that the volume, 𝑉, of a cylinder of radius 𝑟 and height ℎ is given by the formula

𝑉=𝜋𝑟2

Notice that the diagram above shows a cylinder with a height of 13 inches and a diameter of 14 inches. To substitute into the volume formula, we need to know the radius 𝑟, which is half of the diameter. Therefore, our first step is to calculate the radius by dividing the diameter by 2, which gives 𝑟=14÷2=7. We can then substitute this value into the formula, together with ℎ=13, to get

From the question, we need to round our answer to the nearest tenth. The tenths digit is the first digit after the decimal point, which here is a 1. The digit following this (the hundredths digit) is a 9, so our answer must round up to 2‎ ‎001.2 to the nearest tenth.

The diameter and height of the cylinder were given in inches, so the volume is in cubic inches. We conclude that the volume of the cylinder is 2‎ ‎001.2 in3, rounded to the nearest tenth of a cubic inch.

Note that the formula for the volume of a cylinder contains only three variables, 𝑉, 𝑟, and ℎ. This means that we can always work backward to calculate the height or radius of a cylinder if we are given the volume and one of the two other measurements. The next example shows how to rearrange the formula to solve this type of problem.

### Example 3: Finding the Height of a Cylinder given the Volume and Radius

A cylinder has a volume of 900 cm3 and a base with a diameter of 14 cm. Find the height of the cylinder to two decimal places.

Recall that the volume, 𝑉, of a cylinder of radius 𝑟 and height ℎ is given by the formula

𝑉=𝜋𝑟2

In the question, we are given a volume 𝑉=900 and a diameter of 14 and we need to use this information to find the height. Before we can apply the formula, we must work out the radius of the cylinder by dividing the diameter by 2. So, we have 𝑟=14÷2=7. Then, substituting for 𝑉 and 𝑟 in the formula, we have

which is the same as ℎ=5.846…. Rounding this value to two decimal places, we get ℎ=5.85.

As the diameter is given in centimetres, the height will have the same unit of measurement. Thus, the height of the cylinder, rounded to two decimal places, is 5.85 cm.

Notice that in the above example, we substituted the values for 𝑉 and 𝑟, the volume and radius of the cylinder, into the formula 𝑉=𝜋𝑟2ℎ and then rearranged to find the value of the height, ℎ. An alternative strategy is to rearrange the formula to make ℎ the subject and then substitute for 𝑉 and 𝑟 directly. Here, we outline how to derive this formula for the height in terms of the volume and radius.

Starting with the original formula 𝑉=𝜋𝑟2ℎ and rewriting the right-hand side to include multiplication signs, we have

If we were to substitute 𝑉=900 and 𝑟=7 directly into this formula, we would get the same value for the height as obtained in the previous example.

Once you are confident about calculating volumes of cylinders when given a radius or diameter or working backward to calculate the height or radius of a cylinder when given the volume and one of the two other measurements, the next step is to look at some word problems. Often with these questions, you are given a real-life context or an extra level of calculation that you need to complete. Let us look at two examples of this type of problem.

### Example 4: Finding the Amount of Water Needed to Fill a Tank

Given that about 7.5 gallons of water can fill one cubic foot, about how many gallons of water would be in this cylindrical water tank if it was full?

First, we will use the formula to calculate the volume of the cylinder given in the question in cubic feet. Then, since we are told the approximate number of gallons of water that fill one cubic foot, we will multiply the volume of the cylinder by the number of gallons per cubic foot to get the total number of gallons in a full tank.

The diagram shows a cylindrical water tank with a diameter of 20 ft and a height of 12 ft. To apply the formula, we need to know the radius 𝑟, so we halve the diameter to get 𝑟=20÷2=10. Substituting for 𝑟 and ℎ in the formula, we get

Note that we have kept this value in its exact form 1200𝜋, as it will be used in our final calculation.

We now need to work out how many gallons of water will fit in the cylinder. The question tells us that approximately 7.5 gallons will fit in one cubic foot. Since we have just worked out that the cylinder has a volume of 1200𝜋 cubic feet, we must multiply these two numbers together:

1200𝜋×7.5=28274.33388….

Rounding this value to the nearest whole number, we have found that the cylindrical tank would hold approximately 28‎ ‎274 gallons of water if it was full.

For the final example, let us consider a situation where we have to compare the volume of a cylinder to the volume of another 3D shape when given their respective dimensions.

### Example 5: Comparing the Volume of a Cube to a Cylinder

Which has the greater volume, a cube whose edges are 4 cm long or a cylinder with a radius of 3 cm and a height of 8 cm?

Recall that the volume, 𝑉, of a cylinder of radius 𝑟 and height ℎ is given by the formula 𝑉=𝜋𝑟ℎ. This question requires us to compare the volume of a cylinder with the volume of a cube.

Now, the volume of a cube, 𝑉cube, can be found by cubing its side length, ℓ. We are told in the question that the cube has edges that are 4 cm long, so ℓ=4. Thus,

ince all lengths in the question are given in centimetres, the volume of the cube is 64 cm3.

In addition, we know that the cylinder has a radius of 3 cm and a height of 8 cm, so substituting 𝑟=3 and ℎ=8 into the formula 𝑉=𝜋𝑟2ℎ, we get

Therefore, the cylinder has a volume of 226.19 cm3 to two decimal places.

Finally, we need to compare the two volumes. Clearly, 226.19>64, so we conclude that the cylinder has the greater volume.

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