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Volume of Prisms

Use the formula V = Bh

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Volume of Prisms

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?

In this concept, you will learn to calculate the volume of different prisms.

Volume

The volume of a solid figure is the measure of how much three-dimensional space it takes up, or holds. Imagine an aquarium. Its length, width, and height determine how much water the tank will hold. If you fill it with water, the amount of water tells the volume of the tank. Volume is measured in cubic units, because you are multiplying three dimensions: length, width, and height.

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

Unit cubes are singular cubes used to represent one unit. When you “fill up” a solid figure with unit cubes, you can see the unit cubes lining up the figure. Then you 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 you count all of these unit cubes, then you can see that you have 24 unit cubes.

You would write your answer for volume as 24 cubic units. Notice that you use cubic units because you have length times width times height.

The formula for volume is a simplified version of the method above. As you saw, you 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. You can use the area formula for rectangles to find the area of the base: \begin{align*}A=lw\end{align*}A=lw. This is the same as counting the number of unit cubes in each row and the number of rows. Once you find the area, you simply multiply it by the height to add the rest of the layers.

The formula for the volume of a rectangular prism is: \begin{align*}V=B \times h\end{align*}V=B×h.

\begin{align*}B\end{align*}B represents the base area of the prism. Remember, a prism can have a base in the shape of any polygon. Therefore, the formula you need to use to find the area of the base will change. But the process stays the same: you 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 look at an example.

Find the volume of the prism below.

First, find the area of the base, \begin{align*}B\end{align*}B.

\begin{align*}\begin{array}{rcl} A &=& l \times w \\ A &=& 16 \times 9 \\ A &=& 144 \end{array}\end{align*}

AAA===l×w16×9144

Next, put the area of the base in the volume formula.

\begin{align*} \begin{array}{rcl} V &=& B \times h \\ V &=& 144 \times 4 \\ V &=& 576 \end{array} \end{align*}
VVV===B×h144×4576

The answer is 576.

The volume of the prism is \begin{align*} 576 \ cm^3\end{align*}576 cm3.

Note: 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.

If you know the volume and the area of the base of a prism, then you can figure out the missing height dimension of the prism.

Let’s look at an 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, find the area of the base, \begin{align*}B\end{align*}B.

\begin{align*}\begin{array}{rcl} A &=& l \times w \\ A &=& 17.2 \times 11.1 \\ A &=& 190.92 \end{array}\end{align*}
AAA===l×w17.2×11.1190.92

Next, use the volume formula to find the height, \begin{align*}h\end{align*}h.

\begin{align*}\begin{array}{rcl} V &=& B \times h \\ 1145.52 &=& 190.92 \times h \\ \frac{1145.52}{190.92} &=& \frac{190.92 \times h}{190.92} \\ h &=& 6 \end{array}\end{align*}
V1145.521145.52190.92h====B×h190.92×h190.92×h190.926

The answer is 6.

The height of the prism is 6 ft.

Examples

Example 1

Earlier, you were given a problem about Carlos and his tank cleaning.

Carlos wants to find the volume of his temporary bathtub fish tank that is 5.5 feet long, 3.3 feet wide and 2.2 feet deep.

First, find the area of the base, \begin{align*}B\end{align*}B.

\begin{align*}\begin{array}{rcl} A &=& l \times w \\ A &=& 5.5 \times 3.3 \\ A &=& 18.15 \end{array}\end{align*}

AAA===l×w5.5×3.318.15

Next, put the area of the base in the volume formula.

\begin{align*}\begin{array}{rcl} V &=& B \times h \\ V &=& 18.15 \times 2.2 \\ V &=& 39.93 \end{array} \end{align*}

VVV===B×h18.15×2.239.93

The answer is 39.93.

The bathtub will hold \begin{align*}39.93 \ ft^3\end{align*}39.93 ft3 of water.

Example 2

What is the volume of the prism below?

. Note that the base is a triangle so you need to use the area of a triangle formula.\begin{align*}B\end{align*}BFirst, find the area of the base,

\begin{align*}\begin{array}{rcl} A &=& \frac{1}{2}bh \\ A &=& \frac{1}{2}(16)(6) \\ A &=& 48 \end{array}\end{align*}
AAA===12bh12(16)(6)48

Next, put the area of the base in the volume formula.

\begin{align*}\begin{array}{rcl} V &=& B \times h \\ V &=& 48 \times 10 \\ V &=& 480 \end{array} \end{align*}
VVV===B×h48×10480

The answer is 480.

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

Example 3

Find the volume of a triangular prism with:

\begin{align*}b = 12 \ in, \ h = 10 \ in, \ H = 15 \ in\end{align*}b=12 in, h=10 in, H=15 in

First, find the area of the base, \begin{align*}B\end{align*}B. Note that the base is a triangle so you need to use the area of a triangle formula.

\begin{align*}\begin{array}{rcl} A &=& \frac{1}{2}bh \\ A &=& \frac{1}{2}(12)(10) \\ A &=& 60 \end{array} \end{align*}
AAA===12bh12(12)(10)60

Next, put the area of the base in the volume formula.

\begin{align*}\begin{array}{rcl} V &=& B \times h \\ V &=& 60 \times 15 \\ V &=& 900 \end{array} \end{align*}
VVV===B×h60×15900

The answer is 900.

The volume of the prism is \begin{align*}900 \ in^3\end{align*}900 in3.

Example 4

Find the volume of a rectangular prism with a length of 8 m, width of 7 m, and a height of 3 meters.

First, find the area of the base, \begin{align*}B\end{align*}B.

\begin{align*} \begin{array}{rcl} A &=& l \times w \\ A &=& 8 \times 7 \\ A &=& 56 \end{array} \end{align*}
AAA===l×w8×756

Next, put the area of the base in the volume formula.

\begin{align*}\begin{array}{rcl} V &=& B \times h \\ V &=& 56 \times 3 \\ V &=& 168 \end{array}\end{align*}

The answer is 168.

The volume of the prism is \begin{align*}168 \ m^3\end{align*}.

Example 5

Find the volume of a rectangular prism with a length of 10 in, width of 8 in, and a height of 6 inches.

First, find the area of the base, \begin{align*}B\end{align*}.

\begin{align*}\begin{array}{rcl} A &=& l \times w \\ A &=& 10 \times 8 \\ A &=& 180 \end{array}\end{align*}

Next, put the area of the base in the volume formula.

\begin{align*}\begin{array}{rcl} V &=& B \times h \\ V &=& 80 \times 6 \\ V &=& 480 \end{array}\end{align*}

The answer is 480.

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

Review

Look at each figure and then answer the questions that follow.

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?

6. What is the name of the figure pictured above?

7. What is the shape of the bases?

8. What is the shape of the side faces?

9. What is the volume of the figure?

10. What is the name of the figure pictured?

11. What is the volume of the figure?

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

12. 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?

13. 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?

14. True or false. Volume can be used to measure the amount of water in a pool.

15. True or false. Surface area and volume measure the same thing.

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 8.10. 

Resources

Vocabulary

Net

Net

A net is a diagram that shows a “flattened” view of a solid. In a net, each face and base is shown with all of its dimensions. A net can also serve as a pattern to build a three-dimensional solid.
Rectangular Prism

Rectangular Prism

A rectangular prism is a prism made up of two rectangular bases and four rectangular faces.
Surface Area

Surface Area

Surface area is the total area of all of the surfaces of a three-dimensional object.
Triangular Prism

Triangular Prism

A triangular prism is a prism made up of two triangular bases and three rectangular faces.

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