Volume of a prism derivation

Question

I was wondering if there was any way to formalise how I justify the formula for the volume of a prism.

For a given prism, its volume is given by the area of its cross section multiplied by the length of the prism.

I see this intuitively since the prism can be imagined to be made from infinitesimally many small slices of the cross-section throughout its length.

However, I am wondering if there is a way to formualise this?

Furthermore, a cylinder is said to not be a prism. However, the formula for its volume ($\pi r^2 \times h$) is identical area of the cross section multiplied by the length. Is there any reason for this?

Answer 1

1. Convince yourself that volume of a box with sides $a,b,c$ is $abc$.
2. Then, think about having a square of size $1$ and area $1 \cdot 1$ turn into a box by extending it over the height $h$. The resulting box should have volume $1\cdot 1 \cdot h$, using (1) above.
3. Now think about rescaling the original square to fit the base of the prism. You have to scale in such a way that the resulting area becomes $A$, which is the area of the base. This scales by a factor of $A$, and hence the resulting volume should scale by the same factor, since the height is kept the same. So the final volume is $A\cdot h$.