Why is Archimedes volume and surface area of a sphere important?

Archimedes' discovery of the volume and surface area of a sphere laid foundational principles in geometry and calculus, influencing future mathematical research and applications. His formulas, which state that the volume of a sphere is two-thirds that of the cylinder surrounding it and that the surface area is proportional to the square of its radius, are crucial in fields such as physics, engineering, and astronomy. This work not only advanced mathematical understanding but also demonstrated the power of geometric reasoning and the concept of limits, which are essential in calculus. Archimedes' contributions continue to be relevant in various scientific and practical contexts today.