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.

My first rule is that one trapezoid doesn’t change into a different one just by the orientation being different. That is, just tilting a trapezoid and redrawing it doesn’t make the trapezoids different. We can turn the paper they’re drawn on, or if they’re on the computer screen, we can tilt our heads. This actually means two of my cases — the trapezoid at a jaunty angle, and the trapezoid with the long base on top — are really the same as the standard-issue trapezoid with the long base on bottom and the parallel bases running horizontally.

My second rule is that just length alone doesn’t change one kind of trapezoid into another. That is, if the only difference between one trapezoid and another is that both bases are one inch longer in the second diagram, while the other two legs are both the same lengths and make the same angles with respect to the bases, then they’re still the same shape. This is a subtler similarity than the rotating-angle one mentioned above. Imagine we drew the trapezoid on a sheet of rubber paper. Stretching the rubber sheet out horizontally — or vertically — doesn’t make the figure different, just, longer.

My last rule is that while I’ll allow rectangles and parallelograms as trapezoids, I’m going to count squares as kinds of rectangles, and rhombuses as kinds of parallelograms.

So given those, what different kinds of trapezoids are there to draw?

I have an answer in mind, and I’ll share that when I feel satisfied that I’ve fought enough with Photoshop about illustrating them.

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## Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there.
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