Volume of solids
A box can be filled with unit cubes, but a can, a cone and a ball cannot. How do you find how much any of them holds?
Where you are 7 inside
What this area is about
Volume counts the unit cubes of space a solid fills, in units such as \text{cm}^3. Stacking copies of a base gives the volume of a prism or cylinder, and the other formulas are tied back to it.
How these fit together
Stacking. Volume of prisms and cylinders: V = Bh, base area B times height h. Volume of a cylinder carries it out on a circular base, V = \pi r^2 h.
Fractions of a stack. Pyramids and cones hold one third of the prism or cylinder with the same base and the same height, V = \tfrac13 Bh, upright or leaning, as long as h is the perpendicular height. A sphere of radius r encloses \tfrac43 \pi r^3, two thirds of the cylinder that just contains it.
Why they hold. Cavalieri’s principle: if two solids lie between the same two parallel planes and every plane parallel to them cuts cross sections of equal area, the volumes are equal. Why the volume formulas hold uses it, with dissection and a limit, to justify the cylinder, pyramid, cone and sphere results.
Using them. Composite solids and density add or subtract pieces and turn volume into mass.
A path through it
Prisms and cylinders first. After them come a cylinder alone, pyramids and cones, and Cavalieri’s principle. Composite solids follow pyramids and cones; the derivations follow Cavalieri, the cylinder and pyramids and cones. The sphere needs only what a sphere is, but it leans on the cylinder formula, so it reads best after the cylinder.
After this come the disk method, scale factor and volume and modeling with solids.