Cavalieri & the hat-box

Shear the stack

View

Each coin is a cylinder of radius 2.4 cm and thickness 0.4 cm. Sliding a coin sideways does not change the coin.

The numbers

Cavalieri’s principle. Two solids of the same height whose horizontal cross-sections have equal areas at every level have equal volumes. Here every horizontal cut meets exactly one coin in each stack, and it is the same circle both times.

Pour the water

The cone and the cylinder have the same base radius and the same height. Three coneful’s of water fill the cylinder exactly — not roughly, exactly.

The numbers

V_cone = ⅓πr²h — and the same ⅓ works for every pyramid over the same base.

Where’s the maths? This is mensuration — the branch of geometry that measures lengths, areas and volumes of solids. In school you are handed the volume formulas for a cylinder, a cone and a sphere and told to use them, but where do the ⅓ and the ⁴⁄₃ actually come from? They come from one idea, first written down properly by Bonaventura Cavalieri in 1635 and used by Archimedes almost two thousand years earlier: slice the solids. Imagine cutting a solid into a huge number of paper-thin horizontal slices, like a loaf of bread. The volume is the total of all the slice areas. So if two solids stand on the same table, reach the same height, and every horizontal cut gives the same area in both, they must hold the same amount — whatever shape they are. Scene 1 uses that to build a sphere out of a cylinder with a cone drilled out of it, and the only algebra you need is Pythagoras’ theorem.

Cavalieri’s principle. If two solids have the same height, and every plane parallel to the base cuts them in cross-sections of equal area, then the two solids have equal volume. Scene 2 is the picture to keep in your head: a neat stack of coins and a leaning stack of the same coins. Nothing was added or thrown away, so of course the volume is the same — and at every height the cross-section is the same circle.

Scene 1, the hat-box. On the left is a hemisphere of radius r. On the right is a cylinder of radius r and height r with a cone removed — the cone’s point is at the bottom centre and it opens out to the full circle at the top. Cut both at height h:

• In the hemisphere the cut is a disc. Its radius a is a leg of a right-angled triangle whose hypotenuse is the radius r and whose other leg is h. Pythagoras: a² + h² = r², so a = √(r² − h²) and the area is π(r² − h²).
• In the cylinder the cut is a ring. The outer circle always has radius r. The hole is the cone, and because the cone rises at 45° its radius at height h is exactly h. Area = πr² − πh².

Those two expressions are the same thing written twice. Move the slider and watch the “difference” row stay at 0.00 cm² the whole way from the table to the top of the dome. By Cavalieri the two solids therefore have the same volume, and the right-hand one is easy:

V_hemisphere = πr²·r − ⅓πr²·r = πr³ − ⅓πr³ = ⅔πr³, so a whole sphere is V = 2(πr³ − ⅓πr³) = ⁴⁄₃πr³. That is the formula in your formula list, and now it is not magic. Archimedes was so pleased with this that he asked for a sphere inside a cylinder to be carved on his tombstone.

Scene 3, where the ⅓ comes from. A cone and a cylinder with the same base and the same height are not equal — the cone holds exactly one third. Three conefuls of water fill the cylinder to the brim. The same one third holds for any pyramid: V = ⅓ × base area × height, and Cavalieri is what lets you tilt a leaning pyramid upright without changing its volume, so only the base and the height matter.

Drag to orbit · scroll or pinch to zoom · drag the height slider to move the cut.