the volume of a prism and cylinder
How much water fills an aquarium, how much concrete fills a column, how much soup fills a can — these ask for volume, the amount of three-dimensional space a solid occupies, measured in cubic units. For any prism or cylinder the answer is delightfully uniform: stack up copies of the base.
A prism is a solid with two identical parallel bases joined by flat sides (a triangular prism, a rectangular box, a hexagonal column); a cylinder is the same idea with a circular base. In every case the volume is V = B · h, where B is the area of the base and h is the height (the perpendicular distance between the two bases). Picture the solid as B-worth of area swept straight up through a height h, or as h thin layers each of area B stacked on top of each other. For a cylinder B = pi · r^2, so V = pi · r^2 · h; for a box B = l · w, so V = l · w · h.
The height must be measured perpendicular to the bases, not along a slanted edge. And here is the beautiful part: even if the prism leans over so the top base is shifted sideways (an oblique prism), the volume is still B · h with the same perpendicular height. Cavalieri's principle guarantees it — sliding the layers sideways without changing their areas cannot change the total volume.
A cylindrical tank of radius 2 m and height 3 m holds V = pi · 2^2 · 3 = 12·pi ≈ 37.7 m^3 of water — about 37,700 litres.
V = base area × perpendicular height, for any prism or cylinder, even a leaning one.
Use the perpendicular height between the bases, not a slanted edge; by Cavalieri's principle an oblique prism has the same volume as the upright one with the same base and height.