Area is usually defined in terms of a rectangle (sometimes a square) dating back to the ancient Greeks. Having defined area in terms of a rectangle, then other areas can be found -- particularly the area of a general parallelogram. ( This is often demonstrated by dropping an altitude from one of the vertices, "cutting" off the resulting triangle, and reattaching the triangle on the opposite end of the parallelogram forming a rectangle.)
From the parallelogram we can get other shapes. Every triangle is 1/2 of a parallelogram. (Flip or reflect the triangle and then rotate until a side matches a side in the original configuration).Likewise, any trapezoid is 1/2 of a parallelogram. (Flip or reflect across the midline, then rotate 180 degrees and match up to original configuration)
Then, many areas are found by cutting a region into triangles --e.g. the formula for the area of a regular polygon is derived by cutting the polygon into congruent triangles getting A=1/2ap.
area of a triangle = (1/2)*b*h
area of rectangle = w*h
Area of a circle = pi*r^2
Area of a square = a*a
area of a cone = pi*r*l+pi*r^2
area of a cylinder = 2*pi*r*h+2*pi*r^2
area of a trapizoid = (h/2)(b1+b2)
area of eclipse = pi*r1*r2
These are common shapes we usually encounter. I don't know you still believe in what you said. But this is the normal scenario.
But some times when you solve for areas we assume that it to some lengths or heights or angles to be average value. In this instants we use the factor half. But not always.