The sphere volume formula, (4/3)πr³, often just gets memorized at school and left at that. But why exactly 4/3? Not 3/4, and not simply πr³. The person who first proved this formula was the ancient Greek mathematician Archimedes, and the method he used was so clever that you can still verify it with your own eyes today.
Archimedes' idea was, instead of handling the sphere directly, to build a different shape whose volume you can already know matches the sphere's, and compare the two. Place a hemisphere (a ball cut in half) of radius r side by side with a shape made from a cylinder of radius r and height r, with a cone of radius r carved out of it.
These two shapes look completely different on the outside, but remarkably, slice them both horizontally at the same height above the base, and the cross-sectional areas always come out exactly equal. Slice the hemisphere at height h, and you get a circle with radius √(r²−h²), whose area is π(r²−h²). Slice the cylinder-minus-cone shape at that same height h, and you get a donut shape (a ring) with outer radius r and inner radius h, whose area is πr²−πh². Expand the parentheses and the two expressions turn out to be exactly identical: π(r²−h²) = πr²−πh².
This is where a tool called Cavalieri's principle comes in. It's the principle that if you keep slicing two solids at the same height, starting from the base, and the cross-sectional area is equal at every single height, then the two solids' total volumes must also be equal. It's the same logic as having two stacks of paper sliced extremely thin — if every single layer has exactly the same sheet area, no matter how many sheets are stacked, the two stacks' total volume (height × total area) has no choice but to be the same too.
So the hemisphere's volume equals the cylinder's volume minus the cone's volume. The cylinder's volume is πr²×r = πr³, and the cone's volume is a third of that, (1/3)πr³, so the hemisphere's volume becomes πr³ − (1/3)πr³ = (2/3)πr³. A full sphere is two hemispheres joined together, top and bottom, so the sphere's volume is 2 × (2/3)πr³ = (4/3)πr³. Now you can see exactly where that number 4/3 comes from, completely explained.
Archimedes treasured this proof so dearly that, it's said, he requested that when he died, a sphere fitted perfectly inside a cylinder be carved onto his tomb. A sphere that fits perfectly inside a cylinder occupies exactly 2/3 of that cylinder's volume — and he considered this fact the most beautiful result he discovered in his entire life.
On our activity page, you can change the radius with a slider and confirm, in numbers, that the two solids' volumes always calculate out to be exactly equal, and move the slicing height with a slider to directly confirm that the circle's area and the ring's area match exactly, at any height.