WHY is the area of a trapezoid "1/2(base 1 + base 2) times height"

If you dissect a trapezoid, you might end up with a rectangle and two halves of a triangle.

Diagram: http://img38.imageshack.us/i/hahagn.png/

Looking at the diagram, the area of the rectangle is b1 * h. Now, if you fuse the two triangle halves, the base will be b2 - b1. The height is still h. Solving for the area of this triangle:

A = (1/2)bh
A = (1/2)(b2 - b1)h
A = (b2h - b1h) / 2

Adding the two areas, we get:
TA = [(b2h - b1h) / 2] + b1h
TA = (b2h - b1h + 2b1h) / 2
TA = (b2h + b1h) / 2
TA = h(b1 + b2) / 2 (Standard formula for the area of a trapezoid)

Have a good day. And pardon for the lame-*** diagram.

Ok, 1/2 base1+base2 is the average of the two bases.
Then you multiply it by the height to find the area.
Trapezoids have almost nothing to do with squares except for the fact that it has one pair of parallel sides

Because a mathematician from long ago discovered the formula so now we use it.