Solid Geometry Notebook
Surface area counts the skin of a solid; volume counts what the skin encloses — and the two grow at different rates as a solid scales.
The reference formulas fit on one line each. Cube of edge a: surface 6a², volume a³. Sphere of radius r: surface 4πr², volume (4/3)πr³. Cylinder: volume πr²h. Cone: one-third of the cylinder that encloses it.
The one-third factor for pointed solids is ancient. Democritus is credited with first stating that a cone holds a third of its cylinder; Eudoxus gave the first rigorous proof, and the result sits in Book XII of the Elements. Every pyramid and cone, whatever its base, obeys the same third.
| Solid | Surface area | Volume |
|---|---|---|
| Cube (edge a) | 6a² | a³ |
| Sphere (radius r) | 4πr² | (4/3)πr³ |
| Cylinder (r, height h) | 2πr² + 2πrh | πr²h |
| Cone (r, slant l) | πr² + πrl | (1/3)πr²h |
| Pyramid (base B, slant s) | B + ½·perimeter·s | (1/3)·B·h |
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Scaling separates the two measures. Multiply every length by k and surface area multiplies by k² while volume multiplies by k³. This square-cube law explains why fine dust burns fast, why small animals lose heat quickly, and why doubling a storage tank more than doubles its capacity per unit of sheet steel.
The two measures also answer different questions. Surface area is a net problem — the area of the flat blank that wraps the solid. Volume is a filling problem — how many unit cubes, or litres, the solid holds. Confusing them is the classic error of the subject.
A boundary of intuition was drawn in 1900, when Max Dehn answered Hilbert's third problem: two polygons of equal area can always be dissected into each other, but two polyhedra of equal volume — a cube and a regular tetrahedron, for instance — sometimes cannot. Volume, in three dimensions, is strictly more subtle than area.
Further reading