Cylinder Volume vs Sphere

Cylinder Volume vs Sphere is a comparison calculator that reveals Archimedes' famous discovery: a sphere fills exactly two-thirds of the cylinder that perfectly contains it. This tool exists because the relationship between sphere and cylinder volumes is one of the most elegant results in geometry, and this calculator makes it tangible with real numbers.

When a sphere of radius r sits inside a cylinder with the same radius and height equal to the sphere's diameter (h = 2r), the sphere occupies exactly 2/3 of the cylinder's volume. Archimedes was so proud of this result that he requested it be engraved on his tombstone.

This calculator is used by math students, physics students, packaging engineers optimizing containers, and ball bearing designers calculating clearance volumes.

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.

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.