Let’s Think About It
Credit: Jared Tarbell
License: CC BY-NC 3.0
Tim has a cube with a side length of 4 inches. He has a similar cube with dimensions that are twice the first cube. How does the volume of the larger cube compare to the volume of the smaller cube?
In this concept, you will learn to understand scale relationships of area and volume.
You can compare the scale relationships of distance, area and volume when looking at three–dimensional figures. Some examples of three–dimensional figures include a prism or a pyramid. When you compare different measurements, you will see the proportional relationships between them.
Let’s look at a situation involving volume.
Brooke has a scale model of a warehouse. A storage unit is shaped like a rectangular prism and has the dimensions 4 in. by 3 in. by 6 in. If the scale of the model is , what are the actual dimensions of the storage unit? What is the volume?
First, notice that there are two parts to this problem. The first part is figuring out the actual dimensions given that Brooke has a scale model. The second part is figuring out the volume. Start by using the scale to write a proportion for the length, width, and height.
lengthwidthheight0.5 in2 ft=4 inx ft0.5 in2 ft=3 inx ft0.5 in2 ft=6 inx ft
Next, cross multiply for each dimension.
length width height0.52=4x0.52=3x0.52=6x0.5x=2×4 0.5x=2×3 0.5x=2×60.5x=8 0.5x=6 0.5x=12
Then, divide both sides by 0.5 to solve for
length width height0.5x=8 0.5x=6 0.5x=12 0.5x0.5=80.5 0.5x0.5=60.50.5x0.5=120.5 x=16 x=12 x=24The answers are 16, 12, and 24.
The length of the storage unit is 16 ft, the width is 12 ft, and the height is 24 ft.
Then, you need to calculate the volume of the storage unit.
VVV===l×w×h16×12×244608The answer is 4608.
The volume of the storage unit is
or 4608 cubic feet.
There is a relationship between the area of the base of the prism and the volume of the prism. Let’s take a look at how the area of the base of the prism relates to the volume of the prism using the storage unit problem.
The area of the prism is 192 ft2 or 192 square feet.
Now, to see the relationship between the volume of a prism and the area of a prism, divide the volume by the area.
The answer is 24.
Notice that this is the height of the prism.
Prove that the height of the following prism can be found by using a ratio of volume to area for a prism with a length of 6 inches, a width of 5 inches and a height of 9 inches.
First, calculate the volume of the prism.
Next, calculate the area of the prism.
Then, divide the volume by the area.
The answer is 9.
Notice that this is the height of the prism.
Use the following information for the below questions.
A prism has a length of 16 feet, a width of 12 feet and a height of 18 feet.
Find the volume of the prism.
The answer is 3456.
The volume of the prism is
or 3456 cubic feet.
Find the area of the base.
The answer is 192.
The area of the prism is
or 192 square feet.
Write a ratio comparing the volume to the area of the prism and simplify.
The answer is 18.
The height of the prism is 18 feet.
Remember Tim’s cubes?
Tim has two cubes where the larger one is twice the size of the smaller one. This means that there is a scale factor of 2.
First, find the dimensions of the larger cube.
The smaller cube has a side measure of 4 inches. Since this a cube, the
The larger cube will have a side measure of 4 in×2=8 in. Therefore the length=width = height =8 inches.
Next find the volume of both cubes and compare.
Volume of smaller cubeVolume of larger cubeV=l×w×hV=l×w×hV=4 in×4 in×4 in V=8 in×8 in×8 inV=64 in3 V=512 in3
Then, compare the volume of the larger cube to the smaller cube.
Volume of larger cubeVolume of smaller cubeVolume of larger cubeVolume of smaller cube==512648
The answer is 8.
The volume of the larger cube is 8 times the volume of the smaller cube.
Solve each problem.
1. A cube measures 8 feet on each side. A similar cube has dimensions that are twice as large. How does the volume of the larger cube compare to the volume of the smaller cube? Write a ratio to show the comparison.
2. A cube measures 3 inches on each side. A similar cube has dimensions that are half that of the other cube. How does the volume of the larger cube compare to the volume of the smaller cube? Write a ratio to show the comparison
3. A scale model of a sandbox has dimensions 0.5 inch by 3 inches by 4 inches. If the scale of the model is
, what is the volume of the actual sandbox?
4. A cube measures 5 inches on each side. A similar cube has dimensions that are 3 times as large. How does the volume of the larger cube compare to the volume of the smaller cube? Write a ratio to show the comparison.
5. A shipping box measures 16 inches by 12 inches by 8 inches. A second box has a similar size but each dimension is 14 as long. How does the volume of the second box compare to the volume of the first box?
6. Rina’s fish tank has a volume of 8,000 cubic inches. The dimensions of Ava’s fish tank are all 12 the size of Rina’s. What is the volume of Ava’s fish tank?
7. A prism has a width of 6 feet, a length of 8 feet and a height of 12 feet. What is the volume of the prism?
8. What is the area of the base of this prism?
9. What would the volume be of a prism 14 the size of the one described above?
10. What would the volume be of a prism 12 the size of the one described above?
11. What would the volume be of a prism twice the size of the one described above?
12. What ratio can you use to discover the relationship between volume and area?
13. Which measurement will you find when you simplify this ratio?
14. True or false. You can use scale measurement to find the height of a prism.
15. True or false. You can use scale measurement to find the dimensions of a prism.