Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school
Brian was a geometry teacher through the Teach for America program and started the geometry program at his school
The area formula for a trapezoid is found by making a parallelogram made up of two congruent trapezoids. To do this, copy a trapezoid, rotate the copy 180 degrees, and translate to create a parallelogram. The area of a parallelogram is base times corresponding height; since there are two trapezoids, the area of trapezoids formula must be divided in half. Since the bases are not congruent, they must be summed separately.
In order to find the area of any trapezoid, we have to start with what we know. We know the area of any parallelogram is the base times the corresponding altitude or height. Well, with the trapezoid we only have one pair of parallel bases. So I'm going to label those base 1 and base 2, with different numbers below because they're definitely not going to be congruent. And let's say I knew the height between these two bases. What would be the area? Or how would we calculate teh area in terms of something that we know?
Well, the trick here is to take this trapezoid, duplicate it in your head, rotate it 180 degrees. So what I'm going to do is, I'm going to redraw this trapezoid, although I'm duplicating it, rotating it 180 degrees and translating it and I'm going to draw it over on that side. So what I've done is I've said that we got base 1 down below because we've rotated 180 degrees. Up here is base 2 and we have the same height. So if I kind of look at this, it's pretty clear that we're going to have a parallelogram. Something that we know the area of.
So I'm going to come over here and I'm going to write that the area of our parallelogram which I'm going to abbreviate "para," is equal to 2 times the area of a trapezoid.
Now, why would I say that the area of this parallelogram is equal to 2 times area of trapezoid? Well, how many trapezoids do we have? We have 2 trapezoids. So If I want to find out what the area of the trapezoid is, because I know how to calculate the area of a parallelogram, I'm going to have to divide this equation by 2. So I'm going to say that 2 divided by 2 is 1, and the area of a trapezoid is equal to the area of this parallelogram divided in half. So how can we find the area of this parallelogram?
Well, let's start off by saying what is our base and I'm going to grab a different color marker here, and if we have our height, we have a corresponding base that stretches all the way from end to end. And if this piece is b1 and if this piece is b2, then this whole distance right here is b1 plus b2. So instead of writing area fo the parallelogram, I'm going to write base which is b1 plus b2, I'm going to write that in parentheses, times whatever our height is. And we know what our height is, that's just going to be h. We're given that. And we have to divide this in half and we say that this is equal to the area of our trapezoid.
So that wasn't too bad. All we had to do was say, take your trapezoid, duplicate it, rotate it, translate it. You've created a parallelogram and you know the area formula for a parallelogram.
The trick here was realizing that this base right here or your, either of your bases is b1 plus b2. So the area formula for a trapezoid is the quantity of b1 plus b2, you're going to sum those bases times your height, divided all by 2.