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.

A diagram can do wonders in crystallizing one’s understanding of what a problem is actually asking, and help one get to a solution. This is particularly so when the problem calls on any kind of geometric considerations. But a diagram also encourages the viewer to make assumptions, in particular, that whatever is being talked about will look like *this*. I’m not talking about deliberate deceptions, as with the occasional logic puzzle which proves some fallacious result by letting an illustration which looks plausible but actually contains some error. I just mean diagrams that encourage the making an assumption which might not be true. They can lead work astray.

I’m going to need some terminology to hopefully reduce confusion. A trapezoid has two parallel lines; those two lines we call bases. Typically one’s shorter and one’s longer, so, we have the short base and the long base. There are also two other lines, generally at some kind of odd angle. Those we call the legs.

So I started drawing examples of trapezoids, beginning with what might be almost the generic standard-issue trapezoid. At least, it’s the kind I see when I look up formulas for trapezoid area and its other interesting properties, in case it has any. The long base is on the bottom, as if we’re worried it might tip over if laid down the other way. The short base floats above the long one, and the two non-parallel sides aren’t particularly long compared to the long base. It’s almost the archetypical trapezoid. There’ve even been the occasional efforts to create a convenient little symbol meaning “the trapezoid”, in much the same way a Δ can be used to represent a triangle, and they use trapezoids that look like this.

A particular case — a special case, really — of the trapezoid is called the isosceles trapezoid. In that one the two non-parallel legs have the same length, and they both make the same angles with respect to the parallel lines. If you were to extend the non-parallel lines until they came together, the lines and either of the bases would make an isosceles triangle. The short base is centered just above the long base. If it’s possible to make a trapezoid more iconically trapezoid, this is the way to do it.

But that doesn’t have to be so. For example, there’s no need for the short base to be positioned particularly above the long base. It might extend to the right (or the left) of the short leg, and have the legs stretch out at pretty extreme angles. The legs might even be longer than one or both of the bases.

This one shows off how those unstated assumptions can haunt a proof. When I brought up the subject Chiaroscuro did a nice little demonstration of why the formula for a trapezoid’s area has to be so, and he did it by chopping the trapezoid into a rectangle plus two triangles, then figuring out the rectangle area and figuring out the area of those triangles mooshed together. This works very nicely for the isosceles trapezoid, and for the generic trapezoid described up above. And we can use the same sort of reasoning for this off-kilter trapezoid, but we have to put more work into it. There’s not an obvious rectangle to chop out of it, and if we tried slicing triangles off the edges and fitting them together we’d have the figure fall apart. This can be salvaged, and it’s a good exercise to salvage it, but it means the obvious proof needs to be shored up or, worse, to consider special cases. Breaking up a proof into several special cases is logically fine, and sometimes the easiest way to do it, but the human patience for special cases is finite and we’d like to avoid that if we can.

Kant loathed giving examples. “Examples,” he said, “are the go-cart of judgment.” At least, that’s how Norman Kemp Smith translated it and so that’s how it has ended up imprinted on my brain. Sadly, this delightful nugget is lost in the now-standard Paul Guyer translation which reads, “Thus examples are the leading-strings of the power of judgment, which he who lacks the natural talent for judgment can never do without.”

(The word in dispute there is Gängelwagen; “go-cart” is a somewhat better translation, I think, but will tend to mislead modern readers.)

Just prior to this famous remark is a passage that I think mirrors just what you are saying in this post:

“For as far as the correctness and precision of the insight of the understanding is concerned, examples more usually do it some damage, since they only seldom adequately fulfill the condition of the rule (as casus in terminis) and beyond this often weaken the effort of the understanding to again sufficient insight into rules in the universal and independently of the particular circumstances of experience, and thus in the end accustom us to use those rules more like formulas than like principles” (A 134/B 173-4).

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Your remark is just exactly what I was thinking about this essay. I realized that I hadn’t considered some of the possible trapezoids, particularly the one with the short base far out to the side from the short base, when thinking about Chiaroscuro’s (valid) demonstration of the formula for a particular kind of trapezoid.

Also the grading I have been doing has pointed out where students did not separate the parts which apply to all problems of a kind from the parts which apply to just this particular problem. On my better days I spot those and warn about them in class, but there always seems to be something I didn’t think about before the homeworks come back to me.

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Off topic but, I have to know… since it is 12:11 am here, will my post be first or second???

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I think that WordPress orders response in time by Coordinated Universal Time, which does produce some odd divisions in breaking statistics out by day (though it could be worse than what I get in the Eastern Time Zone), but is at least reasonably unambiguous about ordering.

I wonder if there are other default comment orderings available. Someday I must find out how this works.

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Ahh yes. I’d not considered that a trapezoid might have these special cases with the edges; And.. yeah, that gets a lot more complex. Now I’d have to start thinking of negative triangles, and attacking things from the original formula, and seeing if things held, and it might get a lot less obvious, indeed.

–Chi

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I think it isn’t all that bad, though. If I don’t miss anything there are really only four cases of trapezoids that have to be considered for a proof like yours to work, and only one of them needs a ‘negative rectangle’ to be put into consideration.

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Can’t you always split a trapezoid along a diagonal to make two triangles? The bases of the triangles are the bases of the trapezoid, and their common perpendicular height is that of the trapezoid.

Alternately, the typical trapezoid (not a parallelogram) is the difference of two similar triangles, whose bases are the bases of the trapezoid.

The question about diagrams is very important. An algebraic topologist spoke of turning bad pictures into good proofs.

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You can indeed, and that’s the conclusion of this string of essays. The two-triangle method works out a lot more easily. The difference of similar triangles I worked out also, and it’s where I had started going in class.

I think the working out all the kinds of diagrams has been the most fun of this, despite some frustrations with Photoshop in drawing them. (I could have sworn there was a simple draw-a-polygon tool that worked for non-regular polygons, but I can’t seem to make it present itself.)

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