Does the area of any parellelogram equal the length times width?

Answer 1

Let's draw the parallelogram ABCD in which both pairs of opposite sides are parallel and congruent.
From the vertices A and C draw the altitudes AE and CF respectively to the sides DC and AB of the parallelogram, which separates it into two congruent right triangles AED and CFB, and the rectangular AFCE. So that the area of the parallelogram equals to

2(AAED) + AAFCE
= 2[(DE x AE)/2] + (EC x AE)
= (DE x AE) + (EC x AE)
= (DE + EC)AE
= DC x AE
= base x height

Thus, the area of any parallelogram equals to the product of its base and height.