5.1 Finding Volume

Unit Goals

  • Students find the volume of right rectangular prisms and solid figures composed of two right rectangular prisms.

Section A Goals

  • Describe volume as the space taken up by a solid object.
  • Measure the volume of a rectangular prism by finding the number of unit cubes needed to fill it.
  • Use the layered structure in a rectangular prism to find volume.

Section B Goals

  • Describe the calculations from the previous section as $\text{length} \times \text{width} \times \text{height}$ or $\text{area of the base} \times \text{height}$.
  • Find volume using $\text{length} \times \text{width} \times \text{height}$ or $\text{area of the base} \times \text{height}$.

Section C Goals

  • Find the volume of a figure composed of rectangular prisms.
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Section A: Unit Cubes and Volume

Problem 1

Pre-unit

Practicing Standards:  3.MD.C.7.b

Here is a diagram of the floor in a room.

What is the area of the floor? Explain or show your reasoning.

Solution

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Problem 2

Pre-unit

Practicing Standards:  3.MD.C.7.d

What are the missing side lengths? Explain or show your reasoning.

Solution

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Problem 3

Pre-unit

Practicing Standards:  4.MD.A.1

Which of these units would you use to measure the length of a pencil? Select all that apply.

A: centimeter
B: meter
C: kilometer 
D: inch
E: foot
F: yard
G: mile

Solution

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Problem 4

Pre-unit

Practicing Standards:  3.MD.C.7.d

Find the area of the figure shown here. Explain or show your reasoning.

6-sided shape. 

Solution

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Problem 5

Which has greater volume? Explain or show your reasoning.

AFigure made of 3 rows of cubes. First row, 4 cubes. Second row, 2 cubes. Third row, 1 cube.
BRectangular prism. 2 cubes by 2 cubes by 2 cubes.

Solution

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Problem 6

What is the volume of the figure? Explain or show your reasoning.

Figure made of cubes. Base: 3 cubes by 3 cubes by 3 cubes. Tower of 3 cubes vertically stacked on top of back right cube. 

Solution

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Problem 7

  1. What is the volume of this rectangular prism? Explain or show your reasoning.

    Prism. 3 by 2 by 1 cubes.

  2. What is the volume of this rectangular prism? Explain or show your reasoning. 

    Rectangular prism. 3 cubes by 2 cubes by 4 cubes. 

Solution

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Problem 8

Find the volume of each rectangular prism. Explain or show your reasoning.

  1.  
    Prism. 3 by 1 by 4 cubes.
  2.  
    Rectangular prism. Base, 2 cubes by 3 cubes. Height shown with 3 cubes. 

Solution

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Problem 9

Exploration

Find some objects around you and compare their volume.

  1. List the objects.

  2. Which has the least volume? Which has the greatest?

  3. Pick two of your objects and compare their volume.

Solution

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Problem 10

Exploration

  1. How many different rectangular prisms can you make with 18 cubes? Explain or show your reasoning.

  2. How many different rectangular prisms can you make with 24 cubes? Explain or show your reasoning.

  3. How do the side lengths of each prism compare to one another? What patterns do you notice? Is this pattern true for the rectangular prisms you can make with 36 cubes?

Solution

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Section B: Expressions for Finding Volume

Problem 1

Andre and Clare used different strategies to find the volume of this rectangular prism.

Prism. 8 by 6 by 4 units. 

  1. Andre says the volume of this rectangular prism is \(8 \times 24\) cubic units. Explain or show why Andre is correct.

  2. Clare says the volume of the rectangular prism is \(6 \times 32\) cubic units. Explain or show why Clare is also correct.

Solution

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Problem 2

Which expressions represent the volume of this rectangular prism in cubic units?

Rectangular prism. 6 units by 3 units by 4 units. 

Select all that apply.

A: \(3 \times 4 \times 6\)
B: \(24 \times 12\)
C: \(12 + 12 + 12\)
D: \(24 + 24+ 24\)
E: \(18 \times 4\)

Solution

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Problem 3

A box of milk measures 4 cm by 10 cm by 30 cm. What is its volume in cubic centimeters? Explain or show your reasoning.

Prism. 10 by 4 by 30 centimeters.

Solution

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Problem 4

Exploration

A sugar cube has a volume of about 1 cubic centimeter. About how large of a box would you need to hold 1,000,000 sugar cubes?

Solution

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Problem 5

Exploration

Find some things around the school or house. What unit would you use to measure their volume? Choose one of your objects and estimate its volume.

Solution

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Problem 6

Exploration

An object has volume 36 cubic inches. A box has side lengths 1 foot by 3 inches by 4 inches.

  1. What is the smallest number of these objects that can fit in the box? Explain your reasoning.
  2. What is the largest number of these objects that can fit in the box? Explain your reasoning.

Solution

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Problem 7

Exploration

A container has a volume of 120 cubic inches. 

  1. What could the length, width, and height of the container be? 

  2. Can one of the side lengths be 9 inches? Explain or show your reasoning.

Solution

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Section C: Volume of Solid Figures

Problem 1

What is the volume of this figure? Explain or show your reasoning.

Rectangular prism with additional cubes attached to front.

Solution

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Problem 2

Find the volume of the figure. Explain or show your reasoning.

6-sided rectangular prism.

Solution

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Problem 3

Find the volume of the figure. Explain or show your reasoning.

Two rectangular prisms, one in front of the other. 

Solution

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Problem 4

This is a diagram of a bedroom. What is the volume of the bedroom? Explain or show your reasoning.

Solution

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Problem 5

Exploration

  1. Han says that the volume of this rectangular prism is 50 times as great as a 2-inch cube. Do you agree with Han? Explain or show your reasoning.

    Rectangular prism. 10 inch by 8 inch by 5 inch. 

  2. Han says that he can fit 50 2-inch cubes in this rectangular prism. Do you agree with Han? Explain or show your reasoning.

Solution

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Problem 6

Exploration

Two common sizes of shipping boxes are 5 inches by 6 inches by 15 inches and 11 inches by 6 inches by 15 inches. Which size box would you choose to ship the student workbooks for your math class? Explain or show your reasoning.

Solution

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