Cylinder Volume vs Sphere
A sphere fits perfectly inside a cylinder with the same radius and height equal to the sphere's diameter. The sphere fills exactly 2/3 of that cylinder. Archimedes considered this his greatest discovery. Compare both volumes here with any radius.
Cylinder vs Sphere
Archimedes' 2/3 Ratio
For a sphere of radius r: V_sphere = (4/3)πr³
The smallest cylinder that contains this sphere has radius r and height 2r: V_cylinder = πr² × 2r = 2πr³
The ratio: V_sphere / V_cylinder = (4/3)πr³ / 2πr³ = 2/3
The sphere fills exactly two-thirds of the enclosing cylinder. Archimedes was so proud of this result that he asked for a sphere inscribed in a cylinder to be engraved on his tombstone.
Why 2/3?
At height y above the center, the sphere's cross-section is a circle with radius √(r² − y²). Its area is π(r² − y²).
The cylinder's cross-section at the same height is always πr².
The difference at each height is πy² — which is exactly the area of a cone's cross-section. So:
V_cylinder = V_sphere + V_cone 2πr³ = V_sphere + (2/3)πr³ V_sphere = (4/3)πr³
This elegant proof uses Cavalieri's principle and shows the deep connection between sphere, cylinder, and cone.
Practical Applications
Ball bearings in cylindrical housings: the bearing fills 2/3 of the housing volume, leaving 1/3 for lubrication and clearance.
Packaging: a ball in a cylindrical container wastes 1/3 of the space. This matters for shipping efficiency.
Tank design: for a given radius, a spherical tank holds 2/3 the volume of a cylindrical tank with h = 2r, but a sphere has the minimum surface area for its volume — requiring less material per unit stored.