FormulaProved

Volume and surface area of a sphere (Archimedes)

Statement

A sphere of radius $r$ has volume $V = \dfrac{4}{3}\pi r^3$ and surface area $A = 4\pi r^2$ ; both are equal to $\dfrac{2}{3}$ of the volume $2\pi r^3$ and total surface area $6\pi r^2$ of its circumscribing right circular cylinder of radius $r$ and height $2r$ .

Why is it true?

Slice a hemisphere of radius $r$ and a right circular cylinder of radius $r$ and height $r$ with an inverted cone of radius $r$ and height $r$ hollowed out of it by a horizontal plane at height $z$ above the base. By the Pythagorean theorem, the disk cross-section of the hemisphere has radius $\sqrt{r^2 - z^2}$ and area $\pi(r^2 - z^2)$ , which is identical to the area $\pi r^2 - \pi z^2$ of the ring cross-section of the hollowed cylinder. Because every horizontal slice has the same area, the hemisphere has volume $\pi r^3 - \dfrac{1}{3}\pi r^3 = \dfrac{2}{3}\pi r^3$ , so the full sphere has volume $V = \dfrac{4}{3}\pi r^3$ .

Proof sketch

At height $z \in [-r, r]$ , a plane perpendicular to the axis cuts the sphere of radius $r$ in a disk of radius $\sqrt{r^2 - z^2}$ and area $S(z) = \pi(r^2 - z^2)$ . Comparing this slice by Cavalieri's principle with a cylinder of radius $r$ and height $2r$ from which two cones of base radius $r$ and height $r$ have been removed (or integrating $\int_{-r}^{r} \pi(r^2 - z^2)\,dz$ ) gives $V = 2\pi r^3 - 2\cdot\dfrac{1}{3}\pi r^3 = \dfrac{4}{3}\pi r^3$ . Partitioning the sphere into thin pyramids of height $r$ with bases tiling the surface of area $A$ gives $V = \dfrac{1}{3}Ar$ , whence $A = \dfrac{3V}{r} = 4\pi r^2$ .

Stated by

stated -225Archimedes of Syracuse

Proved by

proved -225Archimedes of Syracuse

Topics that use this theorem

Spheres, cylinders, cones Ancient Mathematics

Related theorems

Cavalieri's principle Pythagorean theorem

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Archimedes (translated by Thomas L. Heath) (1897). The Works of Archimedes (On the Sphere and Cylinder, Book I, Propositions 33–34)
Reviel Netz, William Noel (2007). The Archimedes Codex