I remember reading (but unfortunately, not where) that deriving the volume of a cone requires the use of infinitesimals or the method of exhaustion, which feels surprising to me. In two dimensions, one method of determining the area of a circle is to show it is proportional to its circumscribing square and then define $\pi$ to be four times that ratio. In three dimensions, one can show that three pyramids with square base and height equal to side length can be placed together to fill the unit cube, which implies a constant of proportionality of $1/3$. So intuitively, if one could establish a similarity between cones and pyramids as was done for squares and circles, this would show that the volume of a cone is proportional to the volume of a pyramid. Then if the constant could likewise be determined, we would have the volume of the cone without the usage of infinitesimals.
Years ago I saw a beautiful derivation of Archimedes' formula $1:2:3$ for the volumes of the regular cone, hemisphere, and regular cylinder, respectively (here, regular means $h=r$) in God Created the Integers. The argument hinged on circumscribing a cylinder in a hemisphere and showing the cross-sections between the cone and the outside solids were equal; since the cylinder is a regular prism, the result follows. This proof is ingenious but requires the use of infinitesimals, which is what I'm looking to avoid.