Prove that trapezoids and parallelograms are measurable, and that the formulas for their areas are given by

respectively.

* Proof. * Every trapezoid is measurable since, from geometry, we know it is the union of a rectangle and two right triangles (which are pairwise disjoint, and each of which is measurable by the axioms and by the

previous exercise).

Further, its area is the sum of the areas of the right triangles and the rectangle (since they are pairwise disjoint, their intersection has zero area). To calculate this area we specify the lengths of the two unequal sides of the trapezoid to be

and

. The height is denoted by

. Then, the area of the rectangle is

. The area of the triangles is

and

where

. So, denoting the trapezoid by

, we have

Next, a parallelogram is just a special case of a trapezoid, in which ; hence, by the above formula, and denoting the parallelogram by ,

*Related*