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# Volume of Rectangular Prisms

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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 less 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 Concept is all about figuring out the volume of prisms. Pay close attention and you will learn how to figure this out by the end of the Concept.

### Guidance

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 Concept 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 tells 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 $\text{units}^3$ or cubic units.

Let's look at one.

Find the volume of the following figure using unit cubes.

How many cubes are in this figure? We can see that if we count all the cubes, that we have 48 cubes.

The volume of this prism is 48 cubic units or $\text{units}^3$ .

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 $\times$ the width $\times$ the height.

Here is our formula!!

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.

$V=Bh$

The volume is equal to the $B$ , base area of the prism times the height of the prism.

Let's look at one.

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 $V$ , 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.

$B & = lw\\B & = 16 \times 9\\B & = 144 \ cm^2$

The base area is 144 square centimeters. Now we simply multiply this by the height, which represents the number of layers in the prism.

$V & = Bh\\V & = 144 \times 4\\V & = 576 \ cm^3$

The volume of this rectangular prism is 576 cubic centimeters.

We can work with the same rectangular prism but fill it 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.

$V & = lwh\\V & = (16)(9)(4)\\V & = 576 \ cm^3$

Wow! We got the same answer!

Find the volume of each rectangular prism given the following dimensions.

#### Example A

Length of 10 in, width of 8 in, height of 6 inches

Solution: $480$ $in^3$

#### Example B

Length of 8 m, width of 7 m, height of 3 meters

Solution: $168$ $m^3$

#### Example C

Length of 15 ft, width of 12 ft, height of 11 feet

Solution: $1980$ $ft^3$

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 less 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 $\times$ the width $\times$ the height of both of the boxes.

Let’s do the tall one first.

$V&=lwh\\V&=5(4)(18) \\V&=360 \ cubic \ inches$

Now we can find the volume of the wide, flat box.

$V& =lwh \\V & =(6)(5)(12) \\V & =360 \ cubic \ inches$

“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

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.

### Guided Practice

Here is one for you to try on your own.

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 $B$ .

$B & = lw\\B & = 5.5 \times 3.3\\B & = 18.15 \ ft^2$

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

$V & = Bh\\V & = 18.15 \times 2.2\\V & = 39.93 \ ft^3$

Carla’s bathtub can hold 39.93 cubic feet of water.

### 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

11. Length = 10.5 m, width = 11 m, height = 4 m

12. Length = 12 ft, width = 12 ft, height = 8 ft

13. Length = 16 in, width = 8 in, height = 8 in

14. Length = 12 m, width = 12 m, height = 12 m

15. Length = 24 in, width = 6 in, height = 6 in