Archimedes rigorously applies the Method of Exhaustion, a precursor to integral calculus, to calculate areas and volumes. He uses it to find accurate approximations of π (pi), the area of a circle, and the volume of a sphere.

In his treatise 'Measurement of a Circle', Archimedes calculates a very accurate approximation of π (pi), placing it between 3 10/71 and 3 1/7. This represents a significant improvement over earlier estimations.

Archimedes demonstrates that the surface area of a sphere is four times the area of its greatest circle and that the volume of a sphere is two-thirds the volume of a cylinder that circumscribes it. He considered this his most important achievement, requesting a sphere inscribed in a cylinder to be placed on his tomb.

Archimedes studies the spiral that bears his name, defining it mathematically and finding the area of its turns. This work showcases his mastery of geometric reasoning and his ability to analyze complex curves.

In 'The Sand Reckoner', Archimedes addresses the problem of counting very large numbers. He devises a system for naming numbers larger than those in common use, demonstrating the ability to conceive and manipulate extremely large quantities. He uses this to estimate the number of grains of sand required to fill the universe, showcasing his understanding of astronomical dimensions.