Skill: Relate volume to multiplication and addition

Volume tells how much space is inside a rectangular prism. If the prism is filled with unit cubes, its volume is the number of cubes inside:

$$V=\text{length}\times\text{width}\times\text{height}.$$

You can also split a prism into smaller rectangular prisms. Find each smaller volume, then add them. This works because multiplication distributes over addition.

Example: A rectangular prism is $8$ units long, $5$ units wide, and $3$ units high. Find its volume by splitting it into two prisms.

1. Split the length of $8$ units into $5+3$ units.

2. Find the volume of the first prism:

$$5\times 5\times 3=75\text{ cubic units}.$$

3. Find the volume of the second prism:

$$3\times 5\times 3=45\text{ cubic units}.$$

4. Add the two volumes:

$$75+45=120\text{ cubic units}.$$

So, the total volume is

$$\boxed{120\text{ cubic units}}.$$

This matches multiplying the full dimensions at once:

$$8\times 5\times 3=120.$$

The connection is shown by

$$(5+3)\times 5\times 3 =5\times 5\times 3+3\times 5\times 3.$$

The shared dividing face is only a boundary, not extra space, so it is not counted twice.