 
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
.
