# How did the ancient Greeks discover formulas for volume and surface area?

How did the ancient Greeks discover formulas for volume and surface area of different objects, e.g. of a sphere? They did not know about integrals, so there must another way?

• Method of Exhaustion, although the Wikipedia article is not in the same building as correct. – Chappers Jun 9 '15 at 22:09
• And further to that, to find the volume in the first place, Archimedes had the Mechanical Method/Heuristic. It was the loss of this theorem that led, eventually, to the development of the calculus, and hence all of modern mathematics. – Chappers Jun 9 '15 at 22:11
• @Chappers Thanks for the link to the Mechanical Method. So analytical mechanics (at least some part :-) came before the analysis. Amazing. – mvw Jun 9 '15 at 22:14
• @mvw The Greeks had the law of the lever/balance. That's all Archimedes uses. And we have no idea how widespread the Mechanical Method was. That we have it at all is an astonishing concatenation of coincidence. – Chappers Jun 9 '15 at 22:16
• The surface of a sphere was calculated from that of a section of cone. – Rogelio Molina Jun 9 '15 at 22:19

They used exhaustion (approximation) by inner and outer polygons.

• In fact, Archimedes wrote a letter ("The method") to Eratosthenes in order to explain him his method because no one understood how he made his mathematical propositions. – Rubén Ballester Apr 7 '17 at 7:53

Purely speculation.

I would be surprised if the ancient Greeks and Egyptians didn't "stumble upon" their formulae for area and volume by using sand or water. And then later rigorously filling in the details with a proof (by method of exhaustion for example).

What do I mean by sand or water? Make a container that's pretty spherical. Make another cylindrical container that would fairly closely circumscribe the spherical one. Pour water or sand into the spherical one until its full. Then dump that into the cylindrical one. Eyeball it. See that it's about 2/3 of the cylindrical one. Similar things could be attempted with pyramid containers and boxes. With a fairly good notion of what you're setting about to prove, go and prove it (this is an approach to problem-solving that's still used today in research).

We have reason to believe that Greeks used sand to give geometrical demonstrations. And Archimedes seemed to be a fan of water displacement in at least one instance. Don't see how this sort of approach wouldn't have occurred to somebody.

We can create really spherical objects even without high-tech machinery. The National Measurement Institute basically hand-polished a silicon block into the roundest object in the world as a candidate for the international standard for a kilogram. Using a round object, you can cast a mold for a spherical container.