Gabriel's Horn is a solid of revolution commonly featured in calculus textbooks as a counter-intuitive example of a solid having finite volume but infinite surface area. Other examples of solids with surprising geometrical finitude relationships have also appeared in the literature. This article cites several intriguing examples (some of fractal type), adds additional ones, and discusses how these topics can enhance a Calculus II or Real Analysis course by adding cohesiveness between topics, challenging students at multiple levels, and illustrating both the power and limitations of a computer algebra system. (Contains 12 figures.)