Algebra is a language for representing and talking about calculations.
Despite what your teachers might have told you a million times, it isn't *necessary* for everyday life...but neither is art, music, dance, sports, games, history, fiction, or a million other things people do, so...

I bet it's because most of the Western World learned their geometry from the Greeks first, who used the compass and straightedge for their geometry. To make right triangles with a compass, you could set it to a "L" shape...and stand it on it's "legs."
I don't think...

Area = (1/2)(Base)(Height)
Remember that most triangles are 1/2 of a rectangle, so their area is 1/2 of the rectangle's.