Before diving into a battle between the two, we need to better understand their unique background and definitions. Both Surface Area and Volume can be calculated from any three-dimensional shape while only area can be calculated for two-dimensional shapes and is different from the surface area. "The surface area of any given object is the area or region occupied by the surface of the object. Volume is the amount of space available in the object"¯(Raveendran). Surface area is measured in square units and often we look at either the total surface area or the curved surface area. On the other side of things, "volume is the amount of space, measured in cubic units, that an object or substance occupies"¯ (Raveendram). Large lists or tables can be made of the formulas that have been derived for both Surface Area and Volume. Before we jump into the explanation of these formulas and mathematics, let us first cross the old bridge of time to the minds that conjectured and created theorems and ideas that define and solve for Surface Area and Volume.
Archimedes had his hand in the discovery and understanding of Volume and Surface Area. He "calculated quite a large number of values still in use today, including the surface area of a sphere and the volumes of cylinders and cones"¯ (Harding). His findings for how to solve for the volume and surface area came in the likes of circles, ellipses, parabola, hyperbolas, spheres, and cones. These discoveries paved the way for twentieth century mathematicians to expand upon and add to Archimedes' findings.
At the turn of the twentieth century, Henri Lebesgue and Hermann Minkowski played pivotal roles in developing a more effective way to solve for the Surface Area and Volume of different three-dimensional objects. Their work brought a lot of new ideas and concepts known today into fruition. Some of these concepts included ideas like the development of geometric measure theory, which in essence focuses on finding the surface area for irregular objects. Lebesgue's contributions to establishing the theory of surface area and dimensions also allowed for his expansion of the ideas of integration. It was his return to the concepts of length and area, viewing them from a fresh perspective, that thereby provided an alternative definition of the integral¯ (Dunham). It was this work that allowed an increased understanding and ability to describe Surface Area and Volume using integration.
