The synthetic approach to the theory of area and volume initiated by Euclid is described in the first section of this chapter. The area a of a polygonal region P is computed by decomposing P into a finite number of polygonal components with disjoint interiors, which can be reassembled to form a rectangle R with unit base: a is then the altitude of R. The volume of a polyhedral region can be reckoned in a similar way, but for that it is necessary to use some form of Eudoxus's method of exhaustion.