The Story Behind V = 4/3πr³
Over 2,200 years ago, the Greek mathematician Archimedes of Syracuse (c. 287 – c. 212 BC) derived the volume of a sphere without modern algebra or calculus. He considered this theorem his crowning achievement, requesting that a sphere inscribed in a cylinder be engraved upon his tombstone.
Archimedes' Mechanical Method
Archimedes balanced geometric solids on an imaginary lever scale. He proved that if you take a sphere and a cone of equal radius and height, their combined weights balance a cylinder on the other side of the fulcrum.
Volume(Cone) + Volume(Sphere) = Volume(Cylinder)
Since Volume(Cone) = 1/3 π r² (2r) = 2/3 π r³, and Volume(Cylinder) = π r² (2r) = 2 π r³:
Volume(Sphere) = 2 π r³ - 2/3 π r³ = 4/3 π r³.