This chapter is devoted to solving problems related to cylinders, cones, spheres, solids of revolution, figures with curvature, polyhedra, and their combinations. The reader will see how the most important formulas were obtained more than 2000 years ago by the greatest scientist of antiquity, Archimedes, and will learn to find the volumes of three-dimensional figures using Cavalieri’s principle without using integrals or trigonometric substitutions. There is a section on rotating figures which may be of interest to both high school and college students. By using the example of finding the volume of a paraboloid inscribed in a cylinder and then solving problems about a torus, the reader will become familiar with some methods of calculus that clearly follow from the works of the greatest mathematicians of the past.