Archimedes devised a number of geometric formulas that are still used to determine area and volume. He discovered how to figure out the area underneath a parabolic curve, a contribution that would later help Sir Isaac Newton in his development of calculus. Archimedes also found out how to calculate the area of a circle and established a formula for determining the volume of spheres and cylinders.
The Greek letter pi is used to describe the ratio between a circle's circumference and its diameter. Archimedes pinpointed the value of pi to between 3-10/71 and 3-1/7. Archimedes arrived at this figure by inscribing a 96-sided polygon within a circle. His discoveries were published in the work "Measurement of the Circle."
Archimedes famously leapt from his bath and exclaimed "Eureka!" upon making one of his most valuable discoveries, the principle of water displacement. This principle states that a solid body will displace a volume of water equal to that body's own volume. Archimedes also contributed the notion that the buoyant force of an object in a fluid equals the force of gravity on the fluid that is displaced.
Archimedes conceived of the numerical system of exponents, which his era's numerical system did not allow for. He also contributed theories about centers of gravity in solids and plane figures and put this knowledge to practical use, designing military weaponry, mechanical water pumps and lever systems.