How Archimedes Discovered Volume’s Greatest Trick

Imagine you’re curious about how much water fits in a giant urn or how much marble carved away to make a statue. That question is all about volume—the measure of space inside a 3D shape. From cubes to cones, every solid has its own formula. But one of the most surprising stories comes from Archimedes, who figured out a magical relationship between spheres and cylinders more than 2,000 years ago. Let’s unpack his elegant trick and see how it still shapes the world around us.

Where did this come from?

Archimedes lived in ancient Syracuse (now in Sicily) around 250 BCE. He developed the method of exhaustion—a precursor to calculus—to zoom in on curved shapes by slicing them into ever-smaller pieces. His crowning achievement was proving that a sphere’s volume is exactly two-thirds that of its circumscribing cylinder. So proud was he of this result that his tomb was said to feature a sphere inscribed in a cylinder. And yes, that’s also the source of the legend that he shouted "Eureka!" while discovering how to measure irregular volumes with water displacement.

Where you'll see this in real life

Calculating the air supply in scuba tanks (cylinders need precise volume to keep you safe underwater).Determining how much grain a farm silo can hold (farmers use cylinder and cone formulas every harvest).Designing domed stadiums and planetariums (architects use sphere and hemisphere volumes to estimate materials).Formulating medication capsules and food packaging (knowing volume ensures correct dosage and packaging size).

A common misconception

Many students assume that if you double every side of a 3D shape, its volume only doubles too—but volume actually scales with the cube. Double a cube’s edge and its volume jumps from 1 to 8! Remembering that volume grows cubically (not linearly) helps avoid mistakes when scaling models, recipes or even engineering designs.