Solids

A prism is a stack of identical layers, so V = B · h. Cavalieri's principle (1635): if two solids have cross-sections of equal area at every height, they have the same volume. So a skewed stack of coins has the same volume as a straight cylinder.

Project a band of the sphere horizontally onto the cylinder: higher up the sphere is narrower but more steeply tilted — and the two effects cancel exactly. That is why every band of the sphere has area 2πr · w, and the whole sphere 4πr² — the same as the side of the cylinder. The same idea lies behind an equal-area map (Lambert's cylindrical projection).

Archimedes considered the link between the sphere and the cylinder his greatest discovery. According to Cicero, who in 75 BC found the scholar's overgrown tomb in Syracuse, the gravestone was carved with a sphere inscribed in a cylinder.

1 cm³ = 1 mL · 1 dm³ = 1 L · 1 m³ = 1,000 L

Euler's formula: V − E + F = 2

V — vertices, E — edges, F — faces. Leonhard Euler noticed (around 1750) that for every convex polyhedron this combination equals 2. Look at each solid to check the formula.

The formula also works for prisms, pyramids and the truncated solids below — the numbers change, but the result is always 2. Descartes (around 1630) noticed something related: the “angle deficits” (what is missing to 360°) at all the vertices of a convex solid always add up to 720°.

Angles at a vertex

At least three faces meet at a corner of a solid, and their angles must add up to less than 360° — otherwise the faces lie flat or overlap. The faces of a regular solid are identical regular polygons:

This is the argument of Book XIII of Euclid's Elements (around 300 BC), which ends with the construction of the five regular solids and a proof that there are no others.

Solids around us

What can you get from a cube?

A plane cuts a cube along a polygon. Since a cube has 6 faces, and each face contributes at most one side of the cross-section, the possibilities are: a triangle, a quadrilateral (square, rectangle, rhombus, trapezoid…), a pentagon and a hexagon. You will never get a heptagon.

The most beautiful is the regular hexagon: you get it when the plane is perpendicular to a diagonal of the cube and passes through its center. Its vertices are the midpoints of six edges, and for a cube with edge 1 its area is 3√3 / 4 ≈ 1.299.

Conic sections

A cone cut by a plane gives a circle (horizontal cut), an ellipse (slightly tilted), a parabola (plane parallel to the slant side) or a hyperbola (steeper still). The names of these curves were established by Apollonius of Perga in his Conics (around 200 BC). Today we know that planets travel along ellipses, and thrown balls along parabolas (if you ignore air resistance).

The cone and cylinder here are approximated by polyhedra with 128 lateral faces, so the cross-section areas are accurate to a fraction of a percent. A true hyperbola has two branches — the second lies on the “upside-down” cone above the apex.

Pythagoras twice

The base diagonal d_b is the hypotenuse of a triangle with legs a and b. The space diagonal d is the hypotenuse of a second right triangle — with legs d_b and c:

The angle α between the space diagonal and the base: tan α = c / d_b. For a cube d = a√3, and α ≈ 35.26°. The Pythagorean theorem and similar figures in the plane are on the Geometry page.

Doing the math on a can

For a given volume V the height is fixed: h = V / (πr²). Area of metal:

The derivative (see derivative and integral) A′(r) = 4πr − 2V/r² is zero when V = 2πr³, i.e. for h = 2r — the can is as tall as it is wide. For 330 mL this gives r ≈ 3.74 cm and h ≈ 7.49 cm.

Real 330 mL cans are usually about 6.6 cm across and about 11.5 cm tall (with room for foam, a narrowed neck and a domed bottom). Why are they taller? The lid is made of thicker metal than the wall — with lids k times more expensive the optimum is h = 2k·r. A comfortable grip and stacking on shelves also matter, and the curve is flat near the minimum: a few percent more metal costs very little.

The least skin: a sphere

How much surface does it take to enclose 1 liter?

Of all solids with a given volume, the sphere has the smallest surface. That is why soap bubbles and small drops are round (surface tension “pulls” the surface down to a minimum), and why gravity rounds off large planets.

Elephants, mice and ice cubes

When a solid is enlarged k times, its surface grows k² times and its volume k³ times. So a small mouse has a lot of skin relative to its body mass and loses heat quickly — it has to eat a lot. An elephant has the opposite problem: it produces lots of heat but has relatively little skin; its huge ears, rich in blood vessels, help it shed that heat. Crushed ice melts faster because for the same volume it has a much larger surface (similar figures: Geometry).

The Kepler conjecture

Grocers stack oranges so that each one sits in the hollow between four oranges below. The spheres then fill π / (3√2) ≈ 74.05% of space. In 1611 Johannes Kepler wrote, in a little book about snowflakes, that no packing can be denser.

The proof took almost 400 years. Thomas Hales announced it in 1998 (it appeared in print in 2005), but it relied on huge computer calculations that the referees could not fully check. So Hales and his team rewrote the proof so that a computer program could verify it (the Flyspeck project): this was finished in 2014, and the paper was published in 2017.

Marbles poured any old way into a box fill only about 64% of the space — the rest is air.