Explainer: Volumes of Cylinders

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 cuboid. 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, this would 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 cuboid of length 2; thus, the volume would be 2. If we started with two unit cubes side by side, this would have a cross-sectional area of 2 and adding an additional layer we would then have a cuboid with a volume of 4. This can be seen in the shown diagram.

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 21ร—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. So, if we start with a cylinder with a radius of ๐‘Ÿ and height 1, then the area of the cross-section will be ๐œ‹๐‘Ÿ2 square units, and the volume of the cylinder will be ๐œ‹๐‘Ÿ2 cubic units. If we then wanted to calculate the volume of a cylinder of height โ„Ž (โ„Ž layers), we would multiply ๐œ‹๐‘Ÿ2 by โ„Ž: ๐‘‰=๐œ‹๐‘Ÿ2โ„Ž.

This can be seen in the following figure.

Let us look at this in an example.

Example 1: Finding the Volume of a Cylinder given Radius

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

Answer

Firstly, we need to calculate the area of the cross-section of the cylinder. This is a circle so we can use the formula ๐ด=๐œ‹๐‘Ÿ2: ๐ด=๐œ‹ร—(4.2)2=๐œ‹ร—17.64=55.41769441squarefeet.

Notice that we have not yet rounded the area. To calculate the volume, we then need to multiply the cross-sectional area by the height: ๐‘‰=55.41769441ร—6.5=360.2150137cubicfeet.

At this stage, we round our answer to the accuracy required in the question: 360.2 cubic feet.

We can do this directly using the formula ๐‘‰=๐œ‹๐‘Ÿ2โ„Ž: ๐‘‰=๐œ‹ร—(4.2)2ร—6.5โ‰ˆ360.2cubicfeet(1d.p.).

We will also look at an example where we are given the diameter of the cylinder, not the radius. Our approach is very similar 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 Diameter

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

Answer

Here, we have been told the height and the diameter of the cylinder. Firstly, we need to identify the radius of the cylinder which is 14รท2=7. We can then substitute this directly into the formula ๐‘‰=๐œ‹๐‘Ÿ2โ„Ž: ๐‘‰=๐œ‹ร—(7)2ร—13โ‰ˆ2,001.2cubicinches(1d.p.).

What you may be asked to do in a question is leave your answer in terms of ๐œ‹; these questions are usually answered without the aid of a calculator. Let us look at an example.

Example 3: Finding the Volume of a Cylinder Giving Your Answer in Terms of ๐œ‹

The volume of a cylinder is ๐‘‰=๐œ‹๐‘Ÿ2โ„Ž. Find the volume of a cylinder with a radius of 4 cm and a height of 14 cm. Leave your answer in terms of ๐œ‹.

Answer

We need to substitute the values of ๐‘Ÿ and โ„Ž into the formula and then evaluate: ๐‘‰=๐œ‹๏€น42๏…(14)=๐œ‹ร—16ร—14=224๐œ‹cm3.

Note here that we have written 224๐œ‹ and not ๐œ‹224. The former is the standard accepted notation.

When you are confident calculating volumes of cylinders given a radius and diameter and you are happy giving your answer to a specified accuracy, or in terms of ๐œ‹, the next step is to start having a look at some story (or word) problems. Often with these questions, you are given a context or an additional level of calculation that you need to complete. Let us look at two examples of this type of problem.

Example 4: Solving Story Problems Involving the Volume of a Cylinder

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

Answer

Firstly, we need to calculate the volume of the cylinder in cubic feet. We will do this using the standard formula: ๐‘‰=๐œ‹๐‘Ÿ2โ„Ž. Be careful to check whether you are given the radius or the diameter of the cylinder. In this case, we are given the diameter which is 20 ft. So, our radius is 20รท2=10ft. We can now calculate the volume: ๐‘‰=๐œ‹๏€น102๏…(12)=๐œ‹ร—100ร—12=3,769.911184cubicfeet(6d.p.).

We now need to identify how many gallons of water will fit in the cylinder. We are told in the question that about 7.5 gallons will fit in one cubic foot. We have just worked out that the cylinder has a volume of 3,769.91 cubic feet. So, we need to multiply these numbers together: 3,769.911184ร—7.5=28,274.33388.

Therefore, to the nearest whole gallon, the cylinder would hold approximately 28,274 gallons of water.

Example 5: Solving Story Problems Involving the Volume of 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

This question requires us to calculate the volume of a cube, and the volume of a cylinder. A cube is a prism, so we can calculate the volume by finding the area of the cross-section and then multiplying by its length (or depth): ๐‘‰cube=(4ร—4)ร—4=16ร—4=64cm3.

In order to calculate the volume of the cylinder, we will use the standard formula ๐‘‰=๐œ‹๐‘Ÿ2โ„Ž, being sure to check if we have been given the radius or the diameter of the cylinder. In this case, we have the radius so we can substitute into our formula as follows: ๐‘‰cylinder=๐œ‹๏€น32๏…(8)=๐œ‹ร—9ร—8โ‰ˆ226.19cm3(2d.p.).

We now need to compare the sizes of the two volumes. Clearly, 226.19>64; therefore, the cylinder has the greatest volume.

To finish, let us highlight some key points.

Key Points

  1. A cylinder is a type of prism with a circular cross-section.
  2. To calculate the volume of a prism, you find the area of the cross-section and multiply by its length.
  3. The formula for the area of a cylinder is ๐‘‰=๐œ‹๐‘Ÿ2โ„Ž.
  4. Always be careful to check if you are given the radius or the diameter of the cylinder in the question.
  5. When answering story problems, read the question carefully to identify any relevant information that may be useful in answering it.

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