You know that a box's volume is $V = lwh$, even with fractional edges. That rule is part of a bigger one that works for every prism, including prisms with triangle bases.
Key idea
For any right prism, $V = Bh$. $B$ is the area of the base, the face that is the same all the way through the prism, and $h$ is the distance between the two bases.
Why it works
Slice a prism into layers 1 unit thick, parallel to the base. Each layer is a copy of the base, so each holds $B$ cubic units. There are $h$ layers, so the volume is $B \times h$. For a box, the base is a rectangle, $B = lw$, and $V = lwh$ is the same rule.
Worked examples
Worked example: A box with a fraction
Find the volume of a box $2\frac{1}{2}$ ft long, 4 ft wide and 3 ft tall.
- $B = 2\frac{1}{2} \times 4 = 10$ square ft.
- $V = Bh = 10 \times 3 = 30$ cubic ft.
Answer: 30 cubic ft
Worked example: A triangular prism
The prism above has triangle bases with base 10 in. and height 6 in. It is 9 in. long. Find its volume.
- The base is the triangle: $B = \frac{1}{2} \times 10 \times 6 = 30$ square in.
- $V = Bh = 30 \times 9 = 270$ cubic in.
Answer: 270 cubic in.
Worked example: Finding a height
A fish tank holds 360 cubic inches. Its base is 9 in. by 8 in. How tall is it?
- $B = 9 \times 8 = 72$, so $72h = 360$.
- $h = 360 \div 72 = 5$ in.
Answer: 5 in.
Common mistake
Using a rectangular side face as the base of a triangular prism. The base is the face that stays the same shape all the way through, here the triangle. Check: in $V = Bh$, $h$ must be the distance between the two bases, which is the prism's length.