Archimedes of Syracuse regarded his discovery of the exact ratio between the volume of a sphere and its circumscribing cylinder as his proudest achievement—so much that he requested it be engraved upon his tombstone.
Table of Contents
Key Equation
Vsphere = ⅔ Vcylinder
The Circumscribed Cylinder Setup
Consider a cylinder that snugly encloses a sphere of radius r. The cylinder has a circular base with radius r, and its height is equal to the diameter of the sphere: h = 2r. The cylinder volume is therefore V = πr²h = πr²(2r) = 2πr³.
Applying the Archimedean Ratio
Archimedes proved using geometric balance and method of exhaustion that the sphere occupies exactly two-thirds of the cylinder’s volume:
Remarkably, this same 2/3 ratio holds true for their surface areas as well!
Vsphere = (2/3) × (2πr³) = (4/3)πr³.Remarkably, this same 2/3 ratio holds true for their surface areas as well!
Frequently Asked Questions
What is the ratio between a cylinder, sphere, and cone?
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For identical radius and height h = 2r, the volume ratio of Cone : Sphere : Cylinder is exactly 1 : 2 : 3.