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# 8.6: Volume of Prisms and Cylinders

Difficulty Level: At Grade Created by: CK-12

## Introduction

Lunch Break

In the midst of all of the painting, Jose and Carmen sent Alicia out to the grocery store to pick up some lunch. Since the three were painting at Jose’s house, his mother had made some bread and soup seemed like an obvious choice.

When Alicia got to the store, she wasn’t sure how much soup to buy. She picked out a yummy looking organic chicken vegetable soup that she was sure everyone would like, but she couldn’t decide between two and three cans.

After standing there for a few minutes, her stomach began grumbling and she decided to go with the three cans.

“If there is any extra, someone at Jose’s will eat it,” she thought to herself.

Alicia bought the three cans of soup and headed back to Jose’s house.

Each can had a diameter of 5.4 inches and a height of 6.7 inches. What is the total volume of soup that Alicia bought?

To figure this problem out, you will need to know about volume. Pay close attention to the work being done in this lesson and by the end of it, you will know how to answer this question.

What You Will Learn

In this lesson you will learn how to do the following tasks.

• Recognize volume of prisms and cylinders as the sum of volumes of layers of unit cubes.
• Find volumes and linear dimensions of right prisms with any polygonal base.
• Find volumes and linear dimensions of cylinders.
• Solve real – world problems involving volume of prisms and cylinders including metric and customary units of capacity.

Teaching Time

I. Recognize Volume of Prisms and Cylinders as the Sum of Volumes of Layers of Unit Cubes

In the past few lessons we have been learning about finding the surface area of solid figures. Surface area is only one of the measurements that can be calculated. Another measurement is volume. In this lesson we will be calculating the volume of solid figures.

What is volume?

Volume of a solid figure is the measure of how much three-dimensional space it takes up or holds.

Imagine a fish aquarium. Its length, width, and height determine how much water the tank will hold. If we fill it with water, the amount of water tells the volume of the tank. We measure volume in cubic units, because we are multiplying three dimensions: length, width, and height.

There are several different ways that we can calculate volume. The first way that we are going to explore is by looking at we can calculate volume using unit cubes.

What are unit cubes?

Unit cubes are singular cubes used to represent one unit. When we “fill up” a solid figure with unit cubes, we can see the unit cubes lining up the figure. Then we can count or calculate the number of unit cubes in the solid. The number of unit cubes in the solid figure is the volume of the figure. Let’s examine what this would look like in the following prism.

Now you can see that this prism holds unit cubes. It has three unit cubes lined up for the length, it has two unit cubes along the side for the width and it has four unit cubes lined up for the height. If we count all of these unit cubes, then we can see that we have 24 unit cubes.

We would write our answer for volume as 24 cubic units. Notice that we use cubic units because we have length times width times height.

We can also use unit cubes to calculate the volume of a cylinder. This one will be a little funnier because a cylinder is circular and unit cubes are squares. This means that since we can’t cut a unit cube into sections, that it will be impossible for us to calculate an accurate measurement for volume. We will be estimating the volume of the cylinder.

You can see that it doesn’t always work to calculate the volume of a figure by counting unit cubes. You can think about this in the case of the cylinder. There is an easier way. We can calculate the volume of any solid figure by using a formula. Let’s begin with prisms.

II. Find Volumes and Linear Dimensions of Right Prisms with Any Polygonal Base

The formula for volume is a simplified version of the method we just learned. As we saw, we used length and width to find the number of unit cubes in the first layer of the figure. This is the same as finding the area of its base. Let’s see how this works.

In this case, the base of the prism is a rectangle. We can use the area formula for rectangles to find the area of the base: A=lw\begin{align*}A = lw\end{align*}. This is the same as counting the number of unit cubes in each row and the number of rows. Once we find the area, we simply multiply it by the height to add the rest of the layers. Therefore the formula for the volume of a rectangular prism is

V=Bh\begin{align*}V = Bh\end{align*}

B\begin{align*}B\end{align*} represents the base area of the prism. Remember, a prism can have a base in the shape of any polygon. Therefore the formula we need to use to find the area of the base will change. But the process stays the same: we find the area of the face that is the base and then multiply this by the height of the prism. Remember, you will need to figure out the area of the base and then multiply it by the height to figure out the volume.

Let’s try it out.

Example

Find the volume of the prism below.

We simply put the values for the length, width, and height in for the appropriate variables in the formula. Then we solve for V\begin{align*}V\end{align*}, volume.

First we find the area of the base. This is the rectangular side on the bottom. Remember, to find the area of a rectangle we multiply the length times the width. This is our first step.

AAA=lw=(16)(9)=144 sq.cm\begin{align*}A&=lw\\ A&=(16)(9)\\ A&=144 \ sq.cm\end{align*}

The area of the base is 144 sq. cm.

But we are not done yet. We need to figure out the volume, so we will need to take the measurement for the area of the base of the figure and multiply it by the height of the figure. We use the following formula to calculate this measurement.

VVV=Bh=144(4)=576 cm3\begin{align*}V&=Bh\\ V&=144(4)\\ V&=576 \ cm^3\end{align*}

The volume of this rectangular prism is 576 cm3\begin{align*}576 \ cm^3\end{align*}. Notice that we used the exponent three to represent the cubic units of the figure. All volume is measured in cubic units, so you will need to use this exponent when working on figuring out the volume of a solid.

Now write this formula down in your notebook. Be sure to make a note that you will need to find the area of the base of the figure and that this could be different depending on the figure.

Example

What is the volume of the prism below?

As we have seen, the volume formula for any prism is V=Bh\begin{align*}V = Bh\end{align*}. First we need to find the base area. Take a look at the prism above. The base is a triangle, so this time we need to use the formula for the area of a triangle, 12bh\begin{align*}\frac{1}{2} bh\end{align*}, to find B\begin{align*}B\end{align*}. The height of the triangle, h\begin{align*}h\end{align*}, is indicated by a dashed line. The base of the triangle, b\begin{align*}b\end{align*}, is the side perpendicular to the height. Remember, we use the height and base measurements for the triangular face, not the height measurement for the whole prism. Look carefully at the image!

Now let’s use the formula for finding the area of a triangle to find the area of base of the triangle.

BBBB=12bh=12(16)(6)=8(6)=48 in.2\begin{align*}B &= \frac{1}{2} bh\\ B &= \frac{1}{2}(16)(6)\\ B &= 8(6)\\ B &= 48 \ in.^2\end{align*}

We now have the base area: 48 square inches.

Next, we simply multiply it by the height of the prism, according to the volume formula.

VVV=Bh=48(10)=480 in.3\begin{align*}V &= Bh\\ V &= 48 (10)\\ V &= 480 \ in.^3\end{align*}

The volume of this triangular prism is 480 in3\begin{align*}480 \ in^3\end{align*}.

We can also use what we have learned to work backwards. If we know the volume and the area of the base of a prism, then we can figure out the height of the prism which would be the missing dimension. Remember that this is the same thing as working out a puzzle!

Example

The base of a rectangular prism with a volume of 1,145.52 cubic feet has sides of 17.2 feet and 11.1 feet. What is the height of the prism?

First, we need to find the area of the base, B\begin{align*}B\end{align*}. We know this is a rectangular prism, so we use the formula B=lw\begin{align*}B = lw\end{align*}.

BBB=lw=17.2(11.1)=190.92 ft2\begin{align*}B &= lw\\ B &= 17.2 (11.1)\\ B &= 190.92 \ ft^2\end{align*}

We can put this into the formula for B\begin{align*}B\end{align*}. We also have been given the volume of the prism, so we put this in for V\begin{align*}V\end{align*}. Then we solve for h\begin{align*}h\end{align*}, the height.

V1,145.521,145.52÷190.926 ft=Bh=190.92h=h=h\begin{align*}V &= Bh\\ 1,145.52 &= 190.92h\\ 1,145.52 \div 190.92 &= h\\ 6 \ ft &= h\end{align*}

The height of the prism is 6 feet.

Now let’s look at how we can use formulas to find the volume of cylinders.

III. Find Volumes and Linear Dimensions of Cylinders

Finding the volume of a cylinder works the same way. We measure the area of the circular base in two dimensions. Then we add the third dimension by multiplying by the height of the cylinder.

We can use the same formula to calculate the volume of cylinders as we did for prisms. First, we find the area of the bottom face. Remember that the formula we use to find the area of a circle is A=πr2\begin{align*}A = \pi r^2\end{align*}. Then, to “stack” the circles to form a cylinder, we simply multiply the area by the height of the cylinder. This gives us the formula for the volume of a cylinder:

V=πr2h\begin{align*}V = \pi r^2 h\end{align*}

This is the same as the prism formula, V=Bh\begin{align*}V = Bh\end{align*}, except that the shape of a cylinder’s base never changes, so we do not use different area formulas to find B\begin{align*}B\end{align*}. We always use the formula for the area of a circle, which is πr2\begin{align*}\pi r^2\end{align*}.

Write this formula down in your notebook.

All we need to know is the radius of the circular faces and the height of the cylinder. We simply put these numbers into the formula and solve for volume, V\begin{align*}V\end{align*}. Let’s give it a try.

Example

Find the volume of the cylinder below.

We have been given all the information we need in order to solve for volume. Let’s put the numbers into the formula.

VVVVV=πr2h=π(5.52)(2.7)=π(30.25)(2.7)=81.68π=256.48 cm3\begin{align*}V &= \pi r^2 h\\ V &=\pi(5.5^2)(2.7)\\ V &=\pi(30.25)(2.7)\\ V &=81.68 \pi\\ V &=256.48 \ cm^3\end{align*}

The volume of this cylinder is 256.48 cubic centimeters.

We can also think about working backwards if we have the volume and one other dimension. Then we can problem solve to figure out the missing dimension.

Example

A cylinder with a radius of 3 inches has a volume of 140.4π\begin{align*}140.4 \pi\end{align*} cubic inches. What is the height of the cylinder?

What is the problem asking us to find? We need to solve for the height of the cylinder. The problem tells us the radius and the volume. This time the volume is written as a function of pi. This is a way of showing a more specific number, rather than approximating with 3.14. We simply put the whole number into the formula for V\begin{align*}V\end{align*} and then solve for h\begin{align*}h\end{align*}, the height.

V140.4π140.4π140.4π÷9π15.6 in.=πr2h=π(32)h=9πh=hDivide both sides by 9π. π cancels out.=h\begin{align*}V &= \pi r^2h\\ 140.4 \pi &= \pi (3^2) h\\ 140.4 \pi &= 9 \pi h\\ 140.4 \pi \div 9 \pi &= h \quad \text{Divide both sides by} \ 9 \pi. \ \pi \ \text{cancels out.}\\ 15.6 \ in. &= h\end{align*}

We used the volume formula to solve for h\begin{align*}h\end{align*} and found that the height of the cylinder is 15.6 inches.

Now let’s look at how we can apply what we have learned to some real – world problems.

IV. Solve Real – World Problems Involving Volume of Prisms and Cylinders Including Metric and Customary Units of Capacity

We can use the methods we have learned to solve real-world problems involving volume. First, be sure you understand what the question is asking. Second, decide what kind of solid figure the problem involves so that you know which formula to use. Let’s practice with a few problems.

Example

Carlos is cleaning out his fish tank, so he filled the bathtub to the rim with water for his fish to swim in while he empties their tank. If the bathtub is 5.5 feet long, 3.3 feet wide, and 2.2 feet deep, how many cubic feet of water can it hold?

First of all, what is the problem asking us to find? We need to find the volume of the bathtub. Is a bathtub a prism or cylinder? It is a rectangular prism, so we’ll need to use the area formula for rectangles to find B\begin{align*}B\end{align*}.

BBB=lw=5.5×3.3=18.15 ft2\begin{align*}B &= lw\\ B &= 5.5 \times 3.3\\ B &= 18.15 \ ft^2\end{align*}

Now we put this value into the volume formula and solve.

VVV=Bh=18.15×2.2=39.93 ft3\begin{align*}V &= Bh\\ V &= 18.15 \times 2.2\\ V &= 39.93 \ ft^3\end{align*}

Carlos’s bathtub can hold 39.93 square feet of water.

Now let’s go back and use what we have learned to solve the problem from the introduction.

## Real-Life Example Completed

Lunch Break

Here is the original problem once again. Reread it and then answer the question at the end.

In the midst of all of the painting, Jose and Carmen sent Alicia out to the grocery store to pick up some lunch. Since the three were painting at Jose’s house, his mother had made some bread and soup seemed like an obvious choice.

When Alicia got to the store, she wasn’t sure how much soup to buy. She picked out a yummy looking organic chicken vegetable soup that she was sure everyone would like, but she couldn’t decide between two and three cans.

After standing there for a few minutes, her stomach began grumbling and she decided to go with the three cans.

“If there is any extra, someone at Jose’s will eat it,” she thought to herself.

Alicia bought the three cans of soup and headed back to Jose’s house.

Each can had a diameter of 5.4 inches and a height of 6.7 inches. What is the total volume of soup that Alicia bought?

Now it is time for you to figure out the volume of soup that was purchased.

Solution to Real – Life Example

What do we need to find?

We need to find the volume of soup that Alicia bought. Keep in mind that she bought 3 cans of soup. We need to find the volume of one can of soup and then multiply this amount by 3 to find the total volume.

What information have we been given?

First, we know that the soup cans are cylinders, so we’ll need to use the volume formula for cylinders. We also know that the height of each can is 6.7 inches. What is the radius? We have only been given the diameter, which is 5.4 inches. Therefore we need to divide by 2 to find the radius.

5.4÷2=2.7\begin{align*}5.4 \div 2 = 2.7\end{align*}

The radius of each can is 2.7 inches. Now we can put this information into the formula and solve for V\begin{align*}V\end{align*}, volume.

VVVVV=πr2h=π(2.72)(6.7)=π(7.29)(6.7)=48.84π=153.36 in.3\begin{align*}V &= \pi r^2h\\ V &= \pi (2.7^2)(6.7)\\ V &=\pi (7.29)(6.7)\\ V &= 48.84 \pi\\ V &= 153.36 \ in.^3\end{align*}

Each can has a volume of 153.36 cubic inches when we approximate pi as 3.14.

But we’re not done yet! Remember, we need to find the total volume of three cans of soup. Therefore we need to multiply the volume of one can by 3.

153.36×3=460.08 in3\begin{align*}153.36 \times 3 = 460.08 \ in^3\end{align*}

Alicia bought a total of 460.08 cubic inches of soup.

## Vocabulary

Here are the vocabulary words that are found in this lesson.

Volume

the amount of water or capacity that a solid figure can hold. Volume is measured in cubic units.

## Time to Practice

Directions: Look at each figure and then answer the questions about each figure.

1. Name the figure pictured above.
2. What is the length of the figure?
3. What is the width of the figure?
4. What is the height of the figure?
5. What is the volume of the figure?

1. Name the figure pictured above.
2. What is the diameter of the figure?
3. What is the radius is the figure?
4. What is the volume of the figure?

1. What is the name of the figure pictured above?
2. What is the shape of the bases?
3. What is the shape of the side faces?
4. What is the volume of the figure?

1. What is the name of the figure pictured?
2. What is the volume of the figure?

Directions: Use what you have learned about volume to solve each problem.

1. A rectangular prism has a base measuring 16.2 by 14.8 feet. If its volume is 2,877.12 cubic feet, what is its height?
2. What is the height of a cylinder whose radius is 12 meters and volume is 1,296π\begin{align*}1,296 \pi\end{align*}?
3. Kelly is using a rectangular container to fill up a bucket of water. The container is 3.8 inches long, 2.5 inches wide, and 7.2 inches tall. If the bucket holds 1,368 cubic inches of water, how many times will Kelly have to fill the cup in order to fill the bucket?
4. The Berryville Aquarium has a shark tank in the shape of a triangular prism. There is only one shark in the tank, so right now the tank is only 23\begin{align*}\frac{2}{3}\end{align*} full. How many cubic feet of water are in the tank?
5. Based on this, how many cubic feet of water would have the tank be completely full?

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