I have a little iPad app for keeping track of how this blog is doing, and I’m even able to use it to compose new entries and make comments. (The entry about the lottery was one of them.) Mostly it provides a way for me to watch the count of unique visits per day, so I can grow neurotic wondering why it’s not higher. But it also provides supplementary data, such as, what search queries have brought people to the site. The “Trapezoid Week” flurry of posts has proved to be very good at bringing in search referrals, with topics like “picture of a trapezoid” or “how do I draw a trapezoid” or “similar triangles trapezoid” bringing literally several people right to me.
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.
I guess this is a good time to give my answer for the challenge of how many different trapezoids there are to draw. At the least it’ll provide an answer to people who seek on Google the answer to how many trapezoids there are to draw. In principle there’s an infinite number that can be drawn, of course, but I wanted to cut down the ways that seem to multiply cases without really being different shapes. For example, rotating a trapezoid doesn’t make it new, and just stretching it out longer in one direction or another shouldn’t. And just enlarging or shrinking the whole thing doesn’t change it. So given that, how many kinds of trapezoids do I see?
All the popular mathematics blogs seem to challenge readers to come up with answers; I might as well try the same, so I can be disheartened by the responses. In a pair of earlier essays I talked about the problem of drawing differently-shaped trapezoids so as to not overlook figures that might be trapezoids just because the intuition focuses on one shape over others.
So how many different shapes of trapezoids are there to draw? Let me lay out some ground rules.
[ I figure this to be the last important bit of Trapezoid Week. ]
I have one last bit of proving the area formula for the trapezoid, and figure this should wrap up all the exposition I mean to do here. I’m starting again with a trapezoid, with the two parallel bases AB and DE drawn so they’re horizontal. I’m making no assumptions about the legs AD and BE; they might be parallel, they might not be. This makes a proof that’s improved from the difference-of-triangles proof, since it works for more shapes. It’ll work naturally for parallelograms and rectangles, whether you want them to be trapezoids or not. It’s also improved from the twin-trapezoid proof, because it requires so much less extra work.
[ More of Trapezoid Week! Here we make finding the area simpler by doubling the number of trapezoids on the screen. ]
Figuring out the area of a trapezoid based on making it the difference between two triangles works all right. “All right” carries with it a sense of inadequacy. The complaints against it are pretty basic. The first is that it doesn’t work for everything which might be called a trapezoid. Maybe we don’t want to consider parallelograms and rectangles to be particular kinds of trapezoids, but, why rule them out if we don’t have to? The second point is the proof is a little convoluted, requiring us to break out of thinking about trapezoids to remember details of similar triangles. It’d be nice if we had a more direct way of proving things.
[ Trapezoid Week continues! ]
Yesterday I set out a diagram, showing off one example of a trapezoid, with which I mean to show one way to get the formula for a trapezoid’s area. The approach being used here is to find two triangles so that the difference in area between the two is the area of the trapezoid. This can often be a convenient way of finding the area of something: find simple shapes to work with so that the area we want is the sum or the difference of these easy areas. Later on I mean to do this area as the sum of simple shapes.
For now, though, I have the trapezoid set up so its area will be the difference of two triangle areas. The area of a triangle is a simple enough formula: it’s one-half the length of the base times the height. We’ll see much of that formula.
(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.
I was ready to go with a little essay about how I ultimately figured out the area of a trapezoid, based on the formula for the area of triangles, when I realized that it was much easier to show this with a diagram. And I had a diagram drawn out pretty well, at least to the limits of my drawing ability and my power to use Photoshop Elements to do the drawing. But then it struck me that there’s a peril in using a diagram when you want to prove anything, and the nature of those perils deserved some attention.
Continue reading “Setting Out To Trap A Zoid”
When I lecture I like to improvise. I prepare notes, of course, the more detailed the more precise I need to be, but my performing instincts are most satisfied when I just go in front of the class with some key points to hit and maybe a few key lines worked out ahead of time. But I did recently make an iconic mistake, repeating the mathematics instructor’s equivalent of the lawyer asking in court a question without already knowing what the answer will be. Improvisation has to be carefully prepared.