Abolish Trapezoids

As I was recovering from a bout of flu contracted after my visit to the Indiana Senate last Wednesday, I had an idea: Abolish the trapezoid.

“Abolish what?” you may say.

This:

Figure 1: A Trapezoid

A trapezoid is a four-sided figure with a pair of parallel sides. If there are two pairs, it is a parellelogram. If there are two pairs at right angles, it is a rectangle. If there are two pairs at right angles and all four sides are equal, it’s a square.

I am teaching parallelograms tomorrow. The students’ homework while I was sick on Friday was to prove that the area of a parallelogram is the base times the height, just like a rectangle. That’s interesting, and the proof is educational. I decided I’ll give them another theorem to prove: that from any parallelogram, you can construct two identical triangles. From there, it’s easy to prove that the area of a triangle is one-half base times height— and, indeed, I think that’s how Euclid did it.

But will I have time, when the textbook also covers trapezoids and the harder formula and proof for their areas? And isn’t it frustrating, when that formula takes longer to explain, and its proof is less educational and pretty, just clunkier? And isn’t it irritating, when outside of a textbook, nobody uses the formula and nobody needs to. You don’t need to, because you can just cut up the trapezoid into triangles and rectangles and use those formulas, rather than remembering an extra, more complicated, formula. And, besides, I’ve never come across a real trapezoid area calculation in my life.

That’s when the idea struck me. Carry my grumpifying one step further. It’s not just the area formula. Who has even used the word “trapezoid” in daily life, or professional life, or academic life? It is purely a math class word. That’s why you asked “Abolish what?”. You learned it in 10th grade math, but you’ve long forgotten it. We don’t use it, and we don’t need it.¹ Instead of “trapezoid”, say, “Quadrilateral with two parallel sides” if you ever feel short on words to describe some awesome new shape.

Figure 2: A Parallelogram. Is it a trapezoid?

There isn’t even agreement as to what “trapezoid” means. It’s a quadrilateral with one pair of parallel sides, to be sure. But could it have *two* pairs of parallel sides— is a parallelogram a special kind of trapezoid? This question is much debated. As Wikipedia’s Trapezoid says,

There is some disagreement whether parallelograms, which have two pairs of parallel sides, should be regarded as trapezoids. Some define a trapezoid as a quadrilateral having only one pair of parallel sides (the exclusive definition), thereby excluding parallelograms. Others define a trapezoid as a quadrilateral with at least one pair of parallel sides (the inclusive definition), making the parallelogram a special type of trapezoid. The latter definition is consistent with its uses in higher mathematics such as calculus.

This disagreement does lead to the one thing I will regret about dropping trapezoids: I’ve used them for a class discussion of definitions. Discussing and arguing over definitions teaches students several things they need pounded into them not just for math education, but knowledge in general:

1. A definition can be anything, as long as its clearly stated.

2. You can’t always use authority to say which definition is best.

3. There isn’t always a “right” definition, but there’s always a “best” definition. You can argue about definitions, and it does matter. Even if it’s unclear which is best, it’s always clear that some definitions are very bad. But you have to say what you are using the word for, or you can’t say which definition is good and which is bad.

In fact, not only is there disagreement over the definition; there isn’t even agreement on which word to use. In America we use “trapezoid”, but in the United Kingdom they use “trapezium”. They’re both bad words. What does “trapezoid” tell you? Nothing. It’s three syllables of hot air. The word “trapeze” comes from it, but that doesn’t help you remember the shape. It’s got nothing to do with traps. The Greek word τραπέζια (trapezia) means “table”, which is pretty good, but it doesn’t carry that connotation in English. So that’s why it’s hard to remember. “Parallelogram” may be beastly long as a word, but at least it contains hints to its meaning, and trips off the tongue with a meter and a beat. “Trapezoid” just lands with an ugly thud.

“Trapezium” is not better. When it lands it doesn’t make a thud, but it bouces back frivolously. When you say, “I just drew a trapezium,” your students must think you are drawing Dr. Seuss animals. ChatGPT said copyright restrictions prevented it from drawing one in that style, but it did give me what you see in Figure 3. Although at least “trapezium” doesn’t make you think about zombie attacks, it does make you think of “trapeze” or “paramecium”, which accounts for all the cilia in Figure 3. Finally, neither of the plurals the British use sounds right. “Trapezia” sounds like “bacteria”, and “trapeziums” has two zz’s in it, for an equally unfortunate outcome.

Figure 3: A Trapezium, from ChatGPT

Reforming the name wouldn’t help. The very concept is ugly and bad. The novel Flatland is about a two-dimensional world inhabited by triangles, squares, pentagons, and so forth. The lower class is composed of isosceles triangles— triangles with sides equal, but the third side different. Occasionally there are uprisings when some irregular triangle with not just one side different, but all three sides different, is born and not killed at birth but allowed to grow up with its psychotic irregularity. The book does not mention trapezoids; they are too unspeakable. A trapezoid would have to be a demon. If it were four sides with no pattern at all, it would not be so ugly, but having two sides parallel gives it just that little bit of symmetry to play up the asymmetry of the rest of it. It’s like how the ugliest science fiction creatures are the ones who are humanoid, not the ones who are entirely alien.

So, let’s get rid of the whole idea of the trapezoid. I’ll tell the student that they should not waste their brain space memorizing what one is. And if I have enough time tomorrow, I shall instead pull up the topic I had intended for last Friday but had to kill because it was too hard for an unprepared substitute (I came down sick too quickly to prepare one): Euclid Proposition 2, Book I, on how to draw a copy of a line starting from a new point using only straight lines and circles.

Figure 2: Euclid Proposition 2, Book I (from Joyce)

Footnotes

1. In this it is like the words “multiplicand” and “subtrahend” and various others that are in my math text, words I had never heard used in many years of reading mathematical economics. They are beloved of textbooks because although math is a hard subject, memorizing terminology is much easier, so teachers who aren’t really smart enough to do math can happily teach it. We see this even more in science. Elementary science classes are full of terminology, but very weak on explanations of how things in nature work. They could at least teach descriptions; terminology disconnected from either theory or evidence is useless as can be.