8.3 SURVIVAL GEOMETRY II 271version of the Principle of Inclusion-Exclusion (Section 6.3).
But how do we compute area? We need just one more axiom, which can serve to
define area:
The area of a rectangle is the product of its base and height.
Long ago, you learned that the area of a triangle is "one-half base times height."
Let's rediscover this, using the axioms above. The first fact we need is that area is
invariant with respect to shearing.
Example 8.3.1 Two parallelograms that share the same base and whose opposite sides
lie on the same line have equal area.
Proof' In the diagram below, parallelograms ABCD and ABEF share the base AB
with opposite sides lying on the same line (which of course is parallel to AB).We use the notation [.] for area. Notice that
[ABCD] = [ABG] + [BCE]-[EGD]'and
[ABEF] = [ABG] + [ADF]-[EGD].It is easy to check that triangles ADF and BCE are congruent (why?), and hence
[ADF] = [BCE]. Thus [ABCD] = [ABEF]. •
Since we know how to find the area of a rectangle, and since we can always "glue"
two copies of a triangle to form a parallelogram, we easily deduce the classic area
formulas below, along with a very important corollary.
8.3.2 The area of a parallelogram is the product of the base and the height (where
"height" is the length of the perpendicular from a vertex opposite the base to the base).
8.3.3 The area of a triangle is one-half of the product of base and height.
8.3.4 If two triangles share a vertex, and the bases opposite this common vertex lie on
the same line, then the ratio of the areas is equal to the ratio of the bases. For example,
if BD = 4 and BC = 15 below, then
4
[ABD] = B[ABe].