or, The World Through Rectangular Glasses
Now that I’m teaching middle school, I find myself wrestling with the sheer number of area formulas that my students need to know (or at least be passingly familiar with). Rectangles, triangles, parallelograms, trapezia…
The logic is this: A handful of geometric figures keep recurring throughout our world. Once you know how to spot them, they’re everywhere, like the Wilhelm Scream. It’s useful to determine the sizes of these shapes effortlessly, via formulas.
That’s all true, so far as it goes. But reducing geometry to formulas alone can lead to tragic misunderstandings, like when a student asked a friend of mine: “Is there a simple way to remember the difference between volume and surface area?” That’s like asking for a simple way to remember the difference between oceans and deserts: You can only confuse them if you have deep misconceptions about each.
So when I teach these formulas, I try to remind myself of an elegant truth: when it comes to area, everything is rectangles.
And yes, I mean everything.
So let’s begin. With rectangles, finding area is a simple matter of multiplication. In each rectangle, you’ve got a little array of squares:
In this case, we have an array of 5 square centimeters by 3 square centimeters. That’s 15 square centimeters in all. Or, more generally: A = bh.
Now, what about triangles?
Well, just draw them inside rectangles.
Notice that we can divide the rectangle into two sides. The triangle fills half of the left sides, and half of the right. So it must be half of the whole area.
Up next: the parallelogram.
Notice that you can chop off one side of the parallelogram, and move it to the other side, giving you…
…a rectangle! So its area is the same as the rectangle’s.
What about the trapezoid (or, as my new British neighbo(u)rs quaintly insist, the trapezium)? This one’s a little trickier.
One method: double the trapezoid, and look at what it makes: a parallelogram!
The area of this whole thing, per our parallelogram formula, is (b1 + b2) x h, but this is two times too big. So we divide by 2 to get our answer.
Alternatively, you can notice that the trapezoid has a little rectangle that fits inside…
…and that the trapezoid, in turn, fits inside a bigger rectangle:
The smaller rectangle has area b1h. The larger has area b2h. And the trapezoid is exactly halfway between them. Hence: the same area formula!
Up next: we go fly a kite!
Deploying our by-now-familiar trick, we can fit the kite inside a larger rectangle, and notice that it fills exactly half:
Hence, our formula: the area is d1d2/2.
What about a rhombus? The same trick!
If you recall that a square is both a special rectangle (because its angles are all equal) and a special rhombus (because its sides are equal), you’ll see that we can find its area in two ways:
Combining these two formulas, we even get a cool result:
(And you don’t need to phone Pythagoras for help with the proof.)
So far we’ve limited ourselves to simple shapes with straight sides. It’s perhaps not surprising that their areas can be found by the clever application of rectangles. But what about something more exotic, something less obviously rectangular?
What about, say… the circle?
First, slice your circle up like a pizza. Then, rearrange the slices in an alternating pattern, to get a shape like this:
It looks kind of like a parallelogram, right? It has a “height” of roughly r (the radius of the circle) and a “base” of roughly πr.
(Note: If you recall, a circle has circumference 2πr. Notice that this base is half the circumference. That’s πr.)
Now, slice it up even finer, and repeat the process. What do you notice?
It looks even more like a parallelogram. And in fact, it’s starting to look like a rectangle.
This is where you draw upon your imagination. Picture slicing the circle up into finer and finer pieces, like a single pizza being shared among the population of the entire world. Then slice it up even finer, so that there’s a slice for every mouse and moose and bacterium on earth. Then slice it up even finer than that!
What will we get?
The finer we slice, the closer we have to a rectangle. And if you could imagine cutting it into “infinite” slices—impossible, but bear with me—you’d get a perfect rectangle.
What would its area be? Well, base times height… which is πr times r… which is πr2. Area formula proved! QED! (That’s Latin for “Game, set, match.”)
Okay. So we’ve boiled the following shapes down to rectangles:
But I promised you that everything was rectangles. And we haven’t covered everything. We haven’t, for example, found the areas of shapes like these:
So, what can we do? Well, we can try to estimate the area of such a shape using one rectangle, but we won’t get very close:
The one on top is too small, and the one on the bottom is too big. And we’ve got no idea how MUCH too big – it isn’t an obvious fraction.
So what do you do when a rectangle fails? Add more rectangles!
Better, but still not great. What about 8 rectangles?
That’s looking closer. And it points towards a pattern: the more skinny rectangles we allow ourselves to use, the closer we get to the true area.
Summarize the whole thing with some equations, and you get an object familiar to calculus students: the integral!
That’s right: the integral. We’re talking about math’s all-powerful engine for finding the area of any shape you can poke an equation at. And the whole thing is nothing but weaponized rectangles.
So, trying to find an area? Put on those rectangular glasses.