2.5. Surfaces and bodies 81 A cylinder over the set D E2 Summary: 2.5. Surfaces and bodies 81 A cylinder over the set D E2 arises when we move the plane, and with it the set D, in a fixed direction to a certain height. The cylinder is the union of all these images of D so arising. The direction of trans- lation is not necessarily orthogonal to the starting plane E2 : we admit skew cylinders too. Particular examples of this construction include skew circular cylinders, and the parallelpiped. A simple application of Cavalieri's principle yields: Theorem 57 (Volume of the cylinder). A cylinder over a measurable base area D with area A and height h has the volume V = Ah. 2.5.3. Euler's polyhedron formula. First we discuss the concept of a polyhedron, though we restrict ourselves here, as in the next section, to convex and compact polyhedra. By a half space in the space E3 we understand the points lying to one side of a hyperplane, including the points of the hyperplane. Half spaces are thus closed subsets of E3 . Collections: Mathematics