You know $V = l imes w imes h$. Grouping the first two factors gives another useful way to see the same formula.
Idea clave
$V = B \times h$, where $B$ is the area of the base and $h$ is the height. The base area is the number of cubes in one layer; the height is the number of layers.
Por qué funciona
$l \times w$ is the area of the bottom rectangle, and it is also the number of unit cubes that sit on it in one layer. Stacking $h$ such layers gives $B \times h$ cubes. Multiplying in a different order, $(l \times w) \times h = l \times (w \times h)$, gives the same volume, so you can stand the box on any face.
Worked examples
Ejemplo resuelto: Base area given
A prism has a base area of 24 square cm and a height of 5 cm. Find its volume.
- $V = B \times h = 24 \times 5$.
- $24 \times 5 = 120$.
Respuesta: 120 cubic centimeters
Ejemplo resuelto: Find the height
A box has a volume of 120 cubic inches and a base area of 20 square inches. How tall is it?
- $V = B \times h$, so $120 = 20 \times h$.
- $120 \div 20 = 6$: there are 6 layers of 20 cubes.
Respuesta: 6 inches
Error común
Multiplying the base area by another side of the base instead of the height. The base area already uses the length and width; the third factor must be the height, the number of layers. Check: name what each number measures before you multiply.