The set of posts about the area of a trapezoid seems to form a nearly coherent enough whole that it seems worthwhile to make a convenient reference point so that people searching for “how do you find the area of a trapezoid in the most convoluted and over-explained way possible?” have convenient access to it all. So, this is the path of that whole discussion.
(It strikes me, this might just as well be Trapezoid Week here. )
Since I did work out the area of a trapezoid starting from the area formula for triangles, and since I was embarrassed to have not seen it sooner, I decide to share it here, where it may do someone some good, particularly if it’s me for next time I teach a class like this. The punch line is known far ahead of time. The trapezoid is a four-sided figure with two sides parallel. The parallel sides have lengths b1 and b2; they’re considered bases. The two bases are an altitude a apart. The area of the trapezoid then is a * (b1 + b2)/2.
I haven’t got done listing kinds of trapezoids, of course. Arguably I’d never be able to finish, since, after all, couldn’t any possible length of the two bases — the parallel lines — and of different lengths of the diagonal legs be imagined? Well, perhaps, although a lot of those kinds are going to look the same. An isoceles trapezoid where the long base is 10 and the short base 8 looks a lot like one where the long base is 11 and the short base 7.5, at least if the bases are the same distance apart. But there are more cases imaginable.