Surface Area of Sphere

Theorem

The surface area $A$ of a sphere whose radius $a$ is given by:

$A = 4 \pi a^2$


Proof


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Historical Note

The was demonstrated by Archimedes

His proof appears as Proposition $34$ in his On the Sphere and Cylinder.


He also discusses this result in his The Method:

From this theorem, to the effect that a sphere is four times as great as the cone with a great circle of the sphere as base and height equal to the radius of the sphere, I conceived the notion that the surface of any sphere is four times as great as a great circle in it; for, judging from the fact that any circle is equal to a triangle with base equal to the circumference and height equal to the radius of the circle, I apprehended that, in like manner, a sphere is equal to a cone with base equal to the surface of the sphere and height equal to the radius.


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 4$: Geometric Formulas: Sphere of Radius $r$: $4.30$
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): sphere
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): sphere
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 7$: Geometric Formulas: Sphere of Radius $r$: $7.30.$
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): sphere
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $1$: Areas and volumes
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $1$: Areas and volumes
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