One of the many great mathematical discoveries of Archimedes was the relationship between the surface area of a cylinder and a sphere. Archimedes discovered that a sphere that has the same diameter as the height and width of the cylinder is 2/3 of the surface area of the cylinder. Archimedes found this so important that he had a sphere inscribed in a cylinder carved onto his tomb.
Archimedes also used a form of math that is very similar to today’s calculus. He used this to try and find the space on the outside of a parabola. Archimedes also tried to calculate the amount of sand that could fit in the universe.
Another one of Archimedes’ greatest mathematical discoveries had to do with volume, and buoyancy. Archimedes is said to have discovered volume measurement by water displacement when he got into a tub and displaced water. He then promptly jumped out of the tub and ran trough the streets of Sicily naked shouting “eureka” which means “I have found it” This discovery of water displacement was also part of the discovery of buoyancy. Archimedes proposed a theory that the amount of buoyant force of a floating object is equal to the weight of the water displaced.
The lever was another one of Archimedes great works. Although Archimedes did not discover, he did a lot of work which advanced the usefulness, and the knowledge of how a lever works. Archimedes discovered the longer the distance from the fulcrum. The easier it is to lift the object. This discovery made lifting many objects much easier, and led to the development of tools like a wheelbarrow. Another Archimedes quote is "give me a place to stand and i will move the earth" this is referring to how useful a lever is. A good example of a lever and a fulcrum being used today is a carpenter pulling a nail out of a board with a hammer.
These are just some of the mathematical discoveries that Archimedes made, and it is obvious how useful and important they are.