10.6: Volume of Prisms
Introduction
Packing Peanuts
One of the tasks that Candice and Trevor have is to fill boxes with packing peanuts when wrapping something that is fragile. The boxes can hold a good amount of packing peanuts. Each day, one of the students goes to the storeroom to get a big bag of packing peanuts. Then they work on filling 5 – 10 boxes half-way full of packing peanuts so that they are ready to go when the time comes.
“I wonder how much one box holds,” Candice asked Trevor one morning.
“I think that would depend on the box. I think that a tall skinny box would hold more than a wide flat box,” Trevor said, challenging her.
“I don’t. Think about it. Tall may seem like it would hold more, but you can spread out the peanuts in the wide flat box. I think the wide flat one will hold more.”
“Let’s test it out,” Candice said. “We can fill them and then go back and count all of the peanuts.”
“There is an easier way than that. We can use the dimensions and figure out the volume of each box.”
Candice is a bit puzzled by that. She thinks that counting all of the peanuts would be a quicker way to work. The two begin a short argument and decide to try out both methods. Candice begins filling boxes and Trevor begins to work it through mathematically.
Here are the dimensions of the two boxes:
The tall box has a length of 5 inches, a width of 4 inches and a height of 18 inches.
The wide flat box has a length of 12 inches, a width of 6 inches, and a height of 5 inches.
Do you have a prediction which one has the greater volume? This lesson is all about figuring out the volume of prisms. Pay close attention and you will understand by the end of the lesson.
What You Will Learn
By the end of this lesson, you will be able to demonstrate the following skills.
- Recognize volume of prisms as the sum of volumes of layers of unit cubes.
- Find volumes of rectangular prisms using formulas.
- Find volumes of triangular prisms using formulas.
- Solve real-world problems involving volumes of prisms.
Teaching Time
I. Recognize Volume of Prisms as the Sum of Volumes of Layers of Unit Cubes
You have heard the world “volume” before in everyday life. We can talk about the volume of water in a pool or in a glass or in a pitcher. This lesson will focus on how we can find the volume of a prism.
A prism is a three-dimensional figure with two congruent parallel bases and rectangular faces for sides. The prism is named by the polygon which makes up its base.
Volume 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 represents the volume of the tank.
We measure volume in cubic units, because we are multiplying three dimensions: length, width, and height.
We will look at several ways to calculate volume. One way is to use unit cubes.
Volume, as we have said, is the amount of space a three-dimensional solid takes up. One way to find the volume of a prism is to consider how many unit cubes it can contain. A unit cube is simply a cube measuring one inch, one centimeter, one foot, or whatever units of measurement we are using, on all sides. Here are some unit cubes.
We use unit cubes as a way to measure the space inside a solid figure, or its volume. We simply count the number of unit cubes that “fit” into the prism. We begin by counting the number of cubes that cover the bottom of the prism, and then count each layer. Let’s see how this works.
How many cubes do you see here? If we count all of the cubes, you will see that we have 24 cubes in this prism.
The volume of this prism is 24 \begin{align*}\text{units}^3\end{align*}
Let’s look at another example.
Example
Find the volume of the following figure using unit cubes.
How many cubes are in this figure? We can see by counting that there are 48. The volume of this prism is 48 cubic units or \begin{align*}\text{units}^3\end{align*}
Did you notice a pattern here?
If you look carefully, you will see that the volume of the rectangular prism is a function of multiplying the length \begin{align*}\times\end{align*}
Here is our formula!!
II. Find Volumes of Rectangular Prisms Using Formulas
We just discovered the formula for finding the volume of a rectangular prism. Now let’s refine that formula a little further. Here is the formula.
\begin{align*}V=Bh\end{align*}
The volume is equal to the \begin{align*}B\end{align*}
Let’s look at an example.
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 \begin{align*}V\end{align*}
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 by the width.
\begin{align*}B & = lw\\
B & = 16 \times 9\\
B & = 144 \ cm^2\end{align*}
The base area is 144 square centimeters. Now we simply multiply this by the height, which represents the number of layers in the prism.
\begin{align*}V & = Bh\\
V & = 144 \times 4\\
V & = 576 \ cm^3\end{align*}
The volume of this rectangular prism is 576 cubic centimeters.
We can calculate the volume of the same rectangular prism with unit cubes too.
You can see that we could count the unit cubes here to find the volume of the rectangular prism. The other option is to multiply the measurements that we see. This would work as well. Let’s try it and see.
\begin{align*}V & = lwh\\
V & = (16)(9)(4)\\
V & = 576 \ cm^3\end{align*}
Wow! We got the same answer!
10I. Lesson Exercises
Find the volume of each rectangular prism given the following dimensions.
- Length of 10 in, width of 8 in, height of 6 inches
- Length of 8 m, width of 7 m, height of 3 meters
- Length of 15 ft, width of 12 ft, height of 11 feet
Take a few minutes to check your answers with a friend.
III. Find Volumes of Triangular Prisms Using Formulas
You have just finished working with rectangular prisms, now we are going to look at volume with triangular prisms.
What is the difference between a rectangular prism and a triangular prism?
A rectangular prism has two parallel faces that are rectangles, then the other faces are rectangles as well. With a triangular prism, the two parallel faces are triangles and then the other faces are still rectangles. Here is a picture of a triangular prism.
We calculate the volume of triangular prisms almost the same way that we find the volume of rectangular prisms. We still use the formula \begin{align*}V = Bh\end{align*}. However, this time the bottom layer of the prism is a triangle, not a rectangle. Therefore we need to use the area formula for a triangle to find \begin{align*}B\end{align*}. Then we can multiply this amount by the height.
Let’s look at an example to see how this works.
Example
What is the volume of this triangular prism?
As we have seen, the volume formula for any prism is \begin{align*}V = Bh\end{align*}. First we need to find the base area. Because the base is a triangle, we need to use the formula for the area of a triangle: \begin{align*}\frac{1}{2} bh\end{align*}. The height of the triangle, \begin{align*}h\end{align*}, is indicated by a dashed line. The base of the triangle, \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!
So there are two things that we need to accomplish, we need to find the area of one of the triangular bases and then we can take that measurement and multiply it with the height of the entire prism.
\begin{align*}V & = Bh\\ B & = \frac{1}{2} bh\\ B & = \frac{1}{2}(16)(6)\\ B & = 48\\ V & = (48)H\\ V & = (48)(10)\\ V & = 480 \ in^3\end{align*}
The volume of this triangular prism is \begin{align*}480 \ inches^3\end{align*}.
10J. Lesson Exercises
Find the volume of the following triangular prisms.
- \begin{align*}b = 12 \ in, \ h = 10 \ in, \ H = 15 \ in\end{align*}
- \begin{align*}b = 7 \ cm, \ h = 5 \ cm, \ H = 9 \ cm\end{align*}
- \begin{align*}b = 4 \ mm, \ h = 3 \ mm, \ H = 5 \ mm\end{align*}
IV. Solve Real-World Problems Involving Volumes of Prisms
We can use the methods we have learned to solve real-world problems involving area. First, be sure you understand what the question is asking. Second, consider whether the item in the problem is a rectangular or a triangular prism so that you know which formula to use. Let’s practice with a few problems.
Example
Carla is cleaning out her fish tank, so she filled the bathtub to the rim with water for her fish to swim in while she empties their tank. If the bathtub is 5.5 feet long, 3.3 feet wide, and 2.2 feet deep, how much 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 bath tub a rectangular or a triangular prism? It is a rectangular prism, so we’ll need to use the area formula for rectangles to find \begin{align*}B\end{align*}.
\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:
\begin{align*}V & = Bh\\ V & = 18.15 \times 2.2\\ V & = 39.93 \ ft^3\end{align*}
Carla’s bathtub can hold 39.93 cubic feet of water.
Example
Every year Jeanie gets a bottle of her favorite perfume for her birthday. The perfume comes in a bottle shaped like a triangular prism. She is worried that she might run out of perfume before her next birthday because the bottle is only half full. How much perfume does she have left?
First, let’s think about what the problem is asking us to find. We need to know how much perfume Jeanie has left. This means we need to find the volume of a full bottle and then divide by 2. Before we can use the volume formula, we also need to decide what kind of prism this is. The problem tells us that the bottle is in the shape of a triangular prism, so we’ll need to use the area formula for triangles to find the base area, \begin{align*}B\end{align*}.
\begin{align*}B & = \frac{1}{2} bh\\ B & = \frac{1}{2} (6) (4)\\ B & = 3 (4)\\ B & = 12 \ cm^2 \end{align*}
The base area is 12 square centimeters. Now we can put this into the volume formula and solve.
\begin{align*}V & = Bh\\ V & = 12 \times 9\\ V & = 108 \ cm^3\end{align*}
Now we know that the volume of the perfume bottle is 108 cubic centimeters. This is the amount a full bottle can contain. Remember, Jeanie’s bottle is only half full. Therefore we need to divide the volume in half:
\begin{align*}108 \ cm^3 \div 2 = 54 \ cm^3\end{align*}
There are 54 cubic centimeters of perfume left in Jeanie’s perfume bottle.
Real–Life Example Completed
Packing Peanuts
Here is the original problem once again. Reread it and then use math with Trevor to find the volume of both boxes.
One of the tasks that Candice and Trevor have is to fill boxes with packing peanuts when wrapping something that is fragile. The boxes can hold a good amount of packing peanuts. Each day, one of the students goes to the storeroom to get a big bag of packing peanuts. Then they work on filling 5 – 10 boxes half-way full of packing peanuts so that they are ready to go when the time comes.
“I wonder how much one box holds,” Candice asked Trevor one morning.
“I think that would depend on the box. I think that a tall skinny box would hold more than a wide flat box,” Trevor said, challenging her.
“I don’t. Think about it. Tall may seem like it would hold more, but you can spread out the peanuts in the wide flat box. I think the wide flat one will hold more.”
“Let’s test it out,” Candice said. “We can fill them and then go back and count all of the peanuts.”
“There is an easier way than that. We can use the dimensions and figure out the volume of each box.”
Candice is a bit puzzled by that. She thinks that counting all of the peanuts would be a quicker way to work. The two begin a short argument and decide to try out both methods. Candice begins filling boxes and Trevor begins to work it through mathematically.
Here are the dimensions of the two boxes:
The tall box has a length of 5 inches, a width of 4 inches and a height of 18 inches.
The wide flat box has a length of 12 inches, a width of 6 inches, and a height of 5 inches.
First, we can figure out the volume of both of the boxes. We can do this simply by multiplying the length \begin{align*}\times\end{align*} the width \begin{align*}\times\end{align*} the height of both of the boxes.
Let’s do the tall one first.
\begin{align*}V&=lwh\\ V&=5(4)(18) \\ V&=360 \ cubic \ inches \end{align*}
Now we can find the volume of the wide, flat box.
\begin{align*}V& =lwh \\ V & =(6)(5)(12) \\ V & =360 \ cubic \ inches \end{align*}
“WOW!” Trevor exclaimed as Candice was still counting.
“What?” Candice asked looking up from one pile of peanuts.
“They have the same volume!!”
Trevor took his piece of paper to show Candice his work. Then he smiled and the two began picking up the packing peanuts. Use arithmetic was definitely faster in this case!
Vocabulary
Here are the vocabulary words found in this lesson.
- Prism
- a three-dimensional solid with two flat parallel polygon bases and rectangular faces. The bases can be any polygon in shape.
- Volume
- the measure of the space inside a solid figure. Volume often is measured in terms of capacity connected with liquid measure.
- Cubic Units
- volume is measured in cubic units because three parts of a solid are being measured, length, width and height.
Technology Integration
- http://www.mathplayground.com/mv_volume_prisms.html – This is a Brightstorm video on how to find the volume of prisms.
Time to Practice
Directions: Find the volume of each rectangular prism. Remember to label your answer in cubic units.
1. Length = 5 in, width = 3 in, height = 4 in
2. Length = 7 m, width = 6 m, height = 5 m
3. Length = 8 cm, width = 4 cm, height = 9 cm
4. Length = 8 cm, width = 4 cm, height = 12 cm
5. Length = 10 ft, width = 5 ft, height = 6 ft
6. Length = 9 m, width = 8 m, height = 11 m
7. Length = 5.5 in, width = 3 in, height = 5 in
8. Length = 6.6 cm, width = 5 cm, height = 7 cm
9. Length = 7 ft, width = 4 ft, height = 6 ft
10. Length = 15 m, width = 8 m, height = 10 m
Directions: Find the volume of each triangular prism. Remember that \begin{align*}h\end{align*} means the height of the triangular base and \begin{align*}H\end{align*} means the height of the whole prism.
11. \begin{align*}b = 6 \ in, \ h = 4 \ in, \ H = 5 \ in\end{align*}
12. \begin{align*}b = 7 \ in, \ h = 5 \ in, \ H = 9 \ in\end{align*}
13. \begin{align*}b = 10 \ m, \ h = 8 \ m, \ H = 9 \ m\end{align*}
14. \begin{align*}b = 12 \ m, \ h = 10 \ m, \ H = 13 \ m\end{align*}
15. \begin{align*}b = 8 \ cm, \ h = 6 \ cm, \ H = 9 \ cm\end{align*}
16. \begin{align*}b = 9 \ cm, \ h = 7 \ cm, \ H = 8 \ cm\end{align*}
17. \begin{align*}b = 5.5 \ mm, \ h = 4 \ mm, \ H = 4 \ mm\end{align*}
18. \begin{align*}b = 11 \ cm, \ h = 9 \ cm, \ H = 8 \ cm\end{align*}
19. \begin{align*}b = 20 \ ft, \ h = 17 \ ft, \ H = 19 \ ft.\end{align*}
20. \begin{align*}b = 22 \ ft, \ h = 19 \ ft, \ H = 17 \ ft.\end{align*}
Directions: Solve each real-world problem involving volume of prisms.
21. What is the volume of the box of cereal that measures \begin{align*}11 \ cm \times 5 \ cm \times 32 \ cm\end{align*}?
22. Kelly is using a rectangular container to fill up a bucket of water. The container is 3 inches long, 2 inches wide, and 7 inches tall. If the bucket holds 504 cubic inches of water, how many times will Kelly have to fill the rectangular container in order to fill the bucket?
23. 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 \begin{align*}\frac{2}{3}\end{align*} full. How many cubic feet of water are in the tank?