Given that every right triangular region is measurable since it can be obtained as the intersection of two rectangles, prove that every triangular region (not necessarily right triangular) is measurable and has area , where is the base and is the altitude (or height) of the triangle.
Proof. From geometry, we know that every triangular region is the union of two-nonintersecting right triangles with one leg equal to the height of the triangular region, and the sum of the other leg from each right triangle equal to the base of the triangular region.
Since every right triangle is measurable (given), their union is measurable (Axiom 2 of area). Further, denoting the two right triangles and , and the triangular region , we have
(Since and are disjoint, .)
Letting the altitude of the triangular region be denoted by , and its base by , we have,