In this chapter, we present an integration method that originates with Fermat and has also appeared in a modern geometric formulation in Roger B. Nelsen’s Proofs Without Words II (2000). In this modern presentation, which is strongly inspired by geometry, we show how Riemann rectangles used to approximate the area under a curve can be transformed into triangles. In this way, a new figure with the same area is obtained, often leading to striking geometric insights. The method relies only on elementary arguments, in particular similarity and Riemann rectangles. Its theory is developed through a sequence of illustrative examples. Finally, we demonstrate how the fan method can be extended to three-dimensional figures, allowing the determination of volumes. This shows that Fermat’s method applies to a broader range of problems than originally anticipated and that it admits a natural generalization to three-dimensional solids, thereby revealing its previously unrecognized potential.