The volume of a 3D 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 unit. By definition, a single unit cube has a volume of 1 cubic unit, written as 1 cubic unit.
The area of a 2D 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 3D solid shape six-sided figure which has six faces that are rectangles.

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. Consider the applet below to find the volume of a rectangular prism that is 2 units tall, 3 units wide, and 5 units long.

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Refer to the following figures and the downloadable asset to fill in the provided charts.

Five groups of connected cubes are shown. The group labeled Figure B has 2 cubes. The group labeled Figure C has 3 cubes. The group labeled Figure D has 4 cubes. The group labeled Figure E has 8 cubes. The group labeled Figure F has 16 cubes.

Investigate

Consider the following questions once you have filled in the two provided charts.

Is there any relationship between the three numbers that you included for the length, width and height in Table 1 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.

Is there any relationship between the area of the base, the height and the volume of a rectangular prism from Table 2? Turn and talk to a partner about the answers you included in the table and any patterns that you noticed.

Answer the questions below after you have completed the activity.

Discussion

Using some combination of the variables defined, 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 = \text{length}, w = \text{width}, h = \text{height}, and V = \text{volume}.

Do you think that there is a second formula that you can write, using some combination of the variables listed 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 = \text{the area of the base}, h = \text{height}, and V = \text{volume}.

Outcomes

6.G.A.2

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas v = lwh and = bhv to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

Investigation: Formula for volume of a rectangular prism

Develop the formula for volume of a rectangular prism

Outcomes

6.G.A.2

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