Formula for the Area of a Trapezoid
Date: 01/14/2006 at 18:01:37 From: Keely Subject: The area of a trapazoid. How do you find the area of a trapezoid? You have a page on area, but you didn't discuss very well how to find the area of a trapezoid. My math teacher told us to picture it as a parallelogram; and this is the formula he gave: (a+b)*1/2 h. I frankly don't understand what he's talking about. He says to imagine a hinge and that you are taking the top and flipping it over and having it look like a parallogram: 20" ___________________________ / \ The "o" represents the "hinge." 6" /_____________________________\o________________________ / \ / /_________________________________\______________________/3" 25" He says you would take (20+25)*3.
Date: 01/14/2006 at 23:38:33 From: Doctor Peterson Subject: Re: The area of a trapazoid. Hi, Keely. I'm not sure which page you looked at, or what you want to know. It sounds like you aren't really asking HOW (you have the formula) but WHY that formula works, and in particular what in the world your teacher is talking about. I wouldn't expect any old page on trapezoid areas to answer that question! We do give other explanations of the formula, such as Finding the Area of Squares and Triangles http://mathforum.org/library/drmath/view/57661.html Trapezoid: Visual Proof of Area Formula http://mathforum.org/library/drmath/view/55006.html Here's a slightly different method related to your teacher's, which I find very helpful: We can put together TWO of the same trapezoid, one upside down, and get a parallelogram: a b +-----+-----------------+ / \ / / \ / / \ / +-----------------+-----+ b a The area of the parallelogram is (a+b)h, where h is the height; so the area of just one trapezoid is half that, or (a+b)h/2. Your teacher's approach uses two halves of the trapezoid, instead of half of two of them; it takes a little more geometrical knowledge to appreciate it. Cut my trapezoid in two along the line halfway up: a +-----+ / \ +-----------o / \ +-----------------+ b Now take the top and flip it over, as if it were attached by a hinge at point o: +-----------o-----------+ / \ / +-----------------+-----+ b a Notice a couple things about this. First, since the two angles at o are supplementary, they will fit together just right when it's flipped over, so that the top edges are now one straight line, and so are the bottom edges. (There's more to say to prove that, but I'm hoping you can see that it at least makes sense.) Similarly, the left and right sides of the new figure are parallel. So we have a parallelogram. Now, the height of the parallelogram is half the height of the original trapezoid, and the base is the sum of the top and bottom of the trapezoid. So its area is A = (a+b)h/2 as before. The picture also shows you, by the way, that the line you cut is half of (a+b). Do you see why? It takes more to convince someone that this really works, but it makes the formula itself very visual: you actually have a+b and h/2 as the base and height. So I can see why your teacher would like it. It's just a little hard to follow at first. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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