Gabriel's Horn is the surface of revolution of y = 1/x for x greater than 1. Surprisingly this has a finite volume, of π cubic units, but an infinitely large surface area!
Gabriel's Horn is also known as Torricelli's Trumpet because it was investigated by the Italian Evangelista Torricelli (1608–1647). As a young man Torricelli studied in Galileo's home at Arcetri, near Florence, and then, upon Galileo's death, succeeded his teacher as mathematician and philosopher for their good friend and patron, the Grand Duke of Tuscany. Torricelli was amazed by the strange property of his mathematical trumpet and tried various ways to avoid the conclusion that a finite volume could be enclosed by a vessel with an infinite surface area. Unfortunately, he lived before calculus came along to explain the apparent paradox in terms of infinitesimals.