Archimedes’ Secret: Why Spheres Outshine Cylinders

Volume is all about the space an object takes up—think of filling a container with water or packing a sports bag. It’s easy to recite formulas for cylinders, cones and spheres, but behind those equations lies a tale as juicy as a Greek drama. Archimedes, the ancient mathematician, unlocked a beautifully simple relationship between spheres and cylinders that still amazes us today.

Where did this come from?

Archimedes spent years studying curved shapes and, around 250 BC, proved that the volume of a sphere is exactly two-thirds that of the smallest cylinder that encloses it. Legend says he was so proud of this result that he asked for a cylinder and sphere to be engraved on his tombstone. His treatise “On the Sphere and Cylinder” is one of the oldest surviving works of math, showing how clever slicing arguments (later called Cavalieri’s principle) can compare volumes without complicated calculus.

Where you'll see this in real life

1. High-pressure storage: Many gas and liquid tanks (like those for liquefied natural gas) are designed as spheres because a sphere encloses the greatest volume for a given surface area, making it structurally efficient under pressure. 2. Food and drink packaging: Cylindrical cans (think soup or soda) balance easy stacking with manufacturing simplicity, while ice-cream cones are conical because the 1/3 base-area-times-height formula gives just enough volume for a scoop without wasting material. 3. Architecture and domes: Spherical domes and arches distribute loads evenly. Geodesic dome homes use many little triangles to approximate a sphere, maximising interior space while using less material. 4. Medical imaging: Tumours in organs are often roughly spherical. Doctors approximate their volume using the sphere formula to monitor growth or plan treatment doses.

A common misconception

You might think formulas like V_cone = 1/3·(base area)·height are arbitrary—but there’s nothing mystical hiding in that 1/3. It comes straight from comparing cross-sections of a cone and cylinder of the same base and height. Slice both shapes horizontally at the same height: you get circles whose areas maintain a constant ratio of 1:1 or 1:3. That consistent ratio across all slices explains why a cone’s total volume is exactly one third of the matching cylinder’s.