Areas and Volumes: Where the Formulas Come From
The school formulas for the circle, the pyramid and the sphere were first proved in ancient Greece. Euclid's Elements shows that a pyramid is one third of its prism; Archimedes showed that a circle equals a right triangle with legs r and C, and that a sphere is two thirds of its cylinder. Both proved such results by squeezing the shape between figures from inside and outside.
Three formulas everyone meets at school:
circle: πr² pyramid: ⅓ · base · height sphere: 4⁄3 πr³
None of them was first written this way. The Greeks compared one shape with another — a circle with a triangle, a pyramid with a prism, a sphere with a cylinder — and proved each comparison. On this site we follow those proofs from their original sources, step by step.
Three proofs to start with:
- The area of a circle (Archimedes, c. 250 BC): a circle has the same area as a right triangle with legs r and C. Proved with polygons inside and outside the circle.
- The volume of a pyramid (Euclid, Elements XII.7, c. 300 BC): two cuts turn a prism into three pyramids with the same volume — so each is one third. The cone is one third of its cylinder for the same reason (XII.10).
- The volume of a sphere (Archimedes): cone, sphere and cylinder of the same width and height are in the ratio 1 : 2 : 3. He found it with a lever and wanted the figure on his tomb.
Explore
- Area of a circle (Archimedes)A circle has the same area as one right triangle — and Archimedes proved it with two impossibilities.
- Volume of a pyramid (Euclid XII.7)Two cuts turn a prism into three pyramids — different shapes, the same volume.
- Volume of a sphere (Archimedes)Cone, sphere and cylinder of the same width and height: 1 : 2 : 3.
Sources
- Euclid, Elements, Book XII — D. E. Joyce, Clark University
- Archimedes, Measurement of a Circle; On the Sphere and Cylinder I; The Method — T. L. Heath, The Works of Archimedes (1897) and The Method of Archimedes (1912)