Deriving volume formulas
Volume formulas for solids like a cylinder (V = pi x r^2 x h) or a pyramid or cone (each one third of the matching prism or cylinder's volume, like V = 1/3 x pi x r^2 x h for a cone) can be derived informally, for example by stacking thin cross-sectional slices (Cavalieri's principle) to build up the total volume, rather than just given as facts to memorize; once you have a formula, including the sphere's (V = 4/3 x pi x r^3, given rather than derived here), you can use it directly to solve real problems, like finding how much a container holds.
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