The formula for the volume of a frustum is considerably easier to derive, and I'm surprised it wasn't here already. If it is, sorry, but I couldn't find it! It's actually more useful than you'd think; for example, I used it recently to determine the mass of a pile of marbles with only a ruler, two styrofoam cups, and some water. So if you're stranded in the desert with only two cups and some water, you can use this formula to accurately find the mass of small objects!
First of all, you need to know the formula for the volume of a cone.
V = 1/3 πr²h
Now, the easiest (and probably only) way to find the volume of a frustum is to subtract the volume of the missing cone from the volume of the original cone. Consider this horrid ASCII art of a frustum, seen from the side:
^ -
/|\ |
/ | \ |
/ | \ |
/ | \ | missing cone; height = x
/ | \ |
/ | \ |
/______.______\ -
/ | r2 \ |
/ | \ | frustum; height = h
/ | \ |
/__________.__________\ -
r1
As you can see, I've put the missing cone back and drawn a line from the top of the full cone to the center of the base. Glancing quickly at these and applying the cone volume formula, the following are evident:
VB = 1/3 πr1²(h + x)
VS = 1/3 πr2²x
...where B is for the Big cone and S is for the Small cone. The volume of our frustum VF is equal to VB - VS, so we can subtract the bottom equation from the top equation and obtain:
VF = VB - VS = 1/3 πr1²(h + x) - 1/3 πr2²x
VF = 1/3 π * (r1²(h + x) - r2²x)
Now what? Obviously we need to do something with x, and this seems to be what throws most people. If you ignore the left part of the cone, you have two triangles that have two identical angles and are thus similar.
^
|\
|*\
| \
x | \
| \
|_ \
- |_|____\
| r2 \
h | \
|_ \
|_|________\
r1
The angle marked with a star and the two right angles are the identical ones. Now, finding x is a simple matter of proportions:
x / r2 = (x + h) / r1
x * r1 = (x + h) * r2
x r1 = x r2 + h r2
x r1 - x r2 = h r2
x = h r2 / (r1 - r2)
We now have an expression for x in terms of the other three variables. Now it's basic algebra:
VF = 1/3 π * (r1²(h + x) - r2²x)
VF = 1/3 π * (r1² h + r1² x - r2² x)
VF = 1/3 π * (r1² h + (r1² - r2²) * x)
VF = 1/3 π * (r1² h + (r1² - r2²) * h r2 / (r1 - r2))
VF = 1/3 πh * (r1² + (r1 - r2)(r1 + r2) * r2 / (r1 - r2))
VF = 1/3 πh * (r1² + (r1 + r2) * r2)
VF = 1/3 πh * (r1² + r1 r2 + r2²)
Which is, as not stated at the top of this writeup so you'd have to read the whole thing to find it, the formula for the volume of a frustum with height h and radii r1 and r2.
QED
This has been another overly detailed formula derivation/explanation by your friendly neighborhood Eevee.