Lesson Explainer: Volumes of Cylinders | Nagwa Lesson Explainer: Volumes of Cylinders | Nagwa

Lesson Explainer: Volumes of Cylinders Mathematics • 8th Grade

In this explainer, we will learn how to calculate volumes of cylinders and solve problems including real-life situations.

In order to find the volume of a cylinder, let us first remind ourselves how to find the volume of a prism. Remember, a prism is a shape with a constant cross section, for example, a rectangular prism. To find the volume of a prism, we need to calculate the area of its cross section and then multiply this by its length. If you think about a single unit cube, it will have a cross-sectional area of 1 square unit and a volume of 1 cubic unit. If we were to put one cube on top of another cube, this would form a rectangular prism of length 2; thus, the volume would be 2. If we started with two unit cubes side by side, they would have a cross-sectional area of 2 and by adding an additional layer, we would then have a rectangular prism with a volume of 4. This can be seen in the diagram shown below.

If we had a prism with a cross-sectional area of 21 square units and a length of 12 units (in other words, 12 layers), this would have a volume of 21×12=252 cubic units. This same process can be applied to work out the volume of any prism.

A cylinder is a prism with a circular cross section. Therefore, if we wanted to work out the volume of a cylinder, first, we would need to calculate the area of the cross section and then multiply by the length (or height) of the cylinder. Now, recall that a circle of radius 𝑟 has an area of 𝜋𝑟. So, if we start with a cylinder with a radius 𝑟 and height 1, then the area of the cross section will be 𝜋𝑟 square units and the volume of the cylinder will be 𝜋𝑟 cubic units. If we then wanted to calculate the volume of a cylinder of height ( layers), we would multiply 𝜋𝑟 by to get 𝜋𝑟. This can be seen in the following figure.

We have thus obtained the formula for the volume of a cylinder.

Formula: Volume of a Cylinder

The volume, 𝑉, of a cylinder of radius 𝑟 and height is given by the formula 𝑉=𝜋𝑟.

Since 𝜋=3.14159 is just a number, then as long as we know the radius and the height of a cylinder, we can always apply this formula to find its volume. Let us look at this in an example.

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.

Answer

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 is given by the formula 𝑉=𝜋𝑟, we can substitute 𝑟=4.2 and =6.5 into the right-hand side to get 𝑉=𝜋×(4.2)×6.5=𝜋×114.66=360.215.

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.

Answer

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

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 =13, to get 𝑉=𝜋×𝑟×=𝜋×7×13=𝜋×637=2001.194.

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 . 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.

Answer

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

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 900=𝜋×7×900=𝜋×49×.

Dividing both sides by 𝜋 and then by 49 gives 5.846=, which is the same as =5.846. Rounding this value to two decimal places, we get =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=𝜋𝑟 and then rearranged to find the value of the height, . An alternative strategy is to rearrange the formula to make 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=𝜋𝑟 and rewriting the right-hand side to include multiplication signs, we have 𝑉=𝜋×𝑟×.

Dividing both sides by 𝜋 and then 𝑟, we get =𝑉𝜋𝑟.

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.

Formula: Height of a Cylinder

The height, , of a cylinder of volume 𝑉 and radius 𝑟 is given by the formula =𝑉𝜋𝑟.

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?

Answer

Recall the formula for the volume, 𝑉, of a cylinder of radius 𝑟 and height : 𝑉=𝜋𝑟.

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 in the formula, we get 𝑉=𝜋×10×12=1200𝜋.

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?

Answer

Recall that the volume, 𝑉, of a cylinder of radius 𝑟 and height is given by the formula 𝑉=𝜋𝑟. 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, . We are told in the question that the cube has edges that are 4 cm long, so =4. Thus, 𝑉==4=4×4×4=64.cube

Since 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 =8 into the formula 𝑉=𝜋𝑟, we get 𝑉=𝜋×3×8=𝜋×72=226.194.

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.

Let us finish by recapping some key concepts from this explainer.

Key Points

  • A cylinder is a type of prism with a circular cross section. To calculate the volume of a prism, you find the area of the cross section and multiply by its length (or height).
  • The volume, 𝑉, of a cylinder of radius 𝑟 and height is given by the formula 𝑉=𝜋𝑟.
  • Always be careful to check if you are given the radius or the diameter of the cylinder in the question.
  • Since the formula contains only three variables, 𝑉, 𝑟, and , 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.
  • We can use the formula to compare cylinders with each other and with cubes and rectangular prisms; this includes real-world scenarios presented as word problems.

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