There are many rectangular prisms that have a length of 4 units and width of 5 units but differing heights. If $h$ represents the height, then the volume $V$ of such a prism is

$$V=20h$$

The equation shows us that the volume of a prism with a base area of 20 square units is a linear function of the height. Because this is a proportional relationship, if the height gets multiplied by a factor of $a$, then the volume is also multiplied by a factor of $a$:

$$V = 20(ah)$$

What happens if we scale *two *dimensions of a prism by a factor of $a$? In this case, the volume gets multiplied by a factor of $a$ twice, or $a^2$.

For example, think about a prism with a length of 4 units, width of 5 units, and height of 6 units. Its volume is 120 cubic units since $4 \boldcdot 5 \boldcdot 6=120$. Now imagine the length and width each get scaled by a factor of $a$, meaning the new prism has a length of $4a$, width of $5a$, and a height of 6. The new volume is $120a^2$ cubic units since $4a\boldcdot 5a \boldcdot 6=120a^2$.

A similar relationship holds for cylinders. Think of a cylinder with a height of 6 and a radius of 5. The volume would be $150\pi$ cubic units since $\pi \boldcdot 5^2 \boldcdot 6 = 150 \pi$. Now, imagine the radius is scaled by a factor of $a$. Then the new volume is $\pi \boldcdot (5a)^2 \boldcdot 6 = \pi \boldcdot 25a^2 \boldcdot 6 $ or $150a^2 \pi$ cubic units. So scaling the radius by a factor of $a$ has the effect of multiplying the volume by $a^2$!

Why does the volume multiply by $a^2$ when only the radius changes? This makes sense if we imagine how scaling the radius changes the base area of the cylinder. As the radius increases, the base area gets larger in two dimensions (the circle gets wider and also taller), while the third dimension of the cylinder, height, stays the same.