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