# Investigation: Develop the formula for volume of a rectangular prism

Lesson

### Objective

• Identify and use a faster way to find the volume of a rectangular prism

### Vocabulary

While you discuss the Investigation activity below with a partner (or with others in your class), be sure to correctly use the vocabulary terms below.

• The volume of a three-dimensional shape is the amount of space that is contained within that shape, measured in cubic units.
• A unit cube is a cube that has a side length of $1$1 unit. By definition, a single unit cube has a volume of $1$1 cubic unit, written as $1$1 cubic unit.
• The area of a two-dimensional shape is the amount of space that is contained within that shape, measured in square units.
• A unit square is a square that has a side length of 1.  Recall, that by definition, a single unit square has an area of 1 square unit.
• A rectangular prism is a three-dimensional solid shape six sided figure which has six-faces that are rectangles.

### Exploration

When we find the volume of a solid shape, we are basically figuring out how many little unit cubes would fit inside the whole space. Just as when we were finding the area of a two-dimensional shape we were calculating how many unit squares would fit inside its boundary.

If the following cube was $1$1 cubic unit, say $1$1 cm3.

A cube with a side length of $1$1 cm.

Then, in this activity, we will begin by calculating the volume of the following solids by counting the number of unit cubes.  The image below demonstrates which part of each diagram should be used to determine each dimension.

The image below shows a rectangular prism with length $5$5 units, breadth $3$3 units, and height $2$2 units. Notice that the measure of each edge corresponds to the number of unit cubes that could be lined up side by side along that edge.

Figure A

 Figure B Figure C Figure D Figure E Figure F

### Procedure

1.  Refer to the diagrams above to complete the table below.

Figure length width height Volume
Figure A 5 3 2 30
Figure B
Figure C
Figure D
Figure E
Figure F

2.  Is there any relationship between the three numbers that you included for the length, width and height in the table and the Volume of the rectangular prism in the given diagram?  Turn and Talk to a partner about the answers that you included in the table and any patterns that you noticed.

3.  Next, consider only the base of the rectangular prisms in each of the diagrams.  The base of the rectangular prism is the rectangular face that is on the bottom.  Count the unit squares in the base in order to determine the area of the Base of the rectangular prism and complete the table below.

Figure Area of the Base height Volume
Figure A
Figure B
Figure C
Figure D
Figure E
Figure F

4.  Is there any relationship between the area of the base, the height and the volume of a rectangular prism? Turn and Talk to a partner about the answers you included in the table and any patterns that you noticed.

5.  Using some combination of the variables defined below, can you write one formula that you could use in order to determine the volume of a rectangular prism more quickly than by counting the unit cubes?

• Let $l=$l= length
• Let $w=$w= width
• Let $h=$h= height
• Let $V=$V= volume

6.  Do you think that there is a second formula that you can write, using some combination of the variables listed below in order to quickly determine the volume of a rectangular prism?   Turn and talk to a partner about the two formulas that you determined.  Be sure to explain what each part of your formula actually represents in a diagram (such as Figure A) during your discussion.

• Let $B=$B= the area of the Base
• Let $h=$h= height
• Let $V=$V= volume

### Outcomes

#### MGSE7.G.3

Describe the two-dimensional figures (cross sections) that result from slicing three- dimensional figures, as in plane sections of right rectangular prisms, right rectangular pyramids, cones, cylinders, and spheres.

#### MGSE7.G.6

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.