For Pi Day 3/14/16: Triangle to Circles, and Back Again

Around this time last year, we heard a lot about 3/14/15 being a once-in-a-century Pi Day, because the date, including the year, matches the first few digits of pi (3.1415…). But the next digit after 3.1415 in the decimal expansion is 9, giving us 3.14159… So pi is a lot closer to 3.1416 than it is to 3.1415, which is why I want to wish you a Happy once-in-a-century Pi Day today, 3/14/16!

Pi fascinates us because it appears in so many unexpected places. Since pi is ultimately about circles (it’s defined, more or less, as the circumference of a circle of diameter 1), what this really means is that circles appear in many unexpected places. I want to show you one.

You need something with a corner (I used a Rubik’s cube), a cable or string of some kind, and two fixed, firm endpoints (I used the ends of a towel rod). Stretch the cable between the two endpoints across the corner. Now move the corner from one side to the other between the two endpoints, keeping the cable stretched across all three, like this:

Question: what’s the shape that the corner traces out? A triangle, a circle, a parabola, an ellipse, something else?

I told you there was a circle coming, so, yes, it turns out to be a circle. (Well, a semicircle, tracing out 180 degrees, or pi!) I find this a little surprising, because this set-up is quite different from a compass, which is based on spinning a fixed length around a center. Here, the length of cable stretched across the corner extends and then contracts as you move the corner from one side to the other. One good reason to try this out physically is that you can really feel the extension and contraction!

Here is an animation that might start to convince you:

The mathematical fact afoot here is Thales’s Theorem, which says that if you put two points on two opposite ends of a circle (so the line between them is a diameter), then the line segments connecting those points with any third point on the circle will meet at a right angle. To give you an idea of why this is true, and play with pi some more, here’s a picture:

The idea here is that if O is the center of the circle, then the line segments OA, OB, and OC all have the same length, because they’re all radii of the circle. This makes the two triangles AOB and BOC isosceles: each triangle has two equal sides, hence two equal angles, as illustrated in the picture. With this notation, the sum of the angles of triangle ABC is $2 \alpha + 2 \beta$. But we know that the sum of angles of a triangle is 180 degrees (pi again!). Dividing by 2, we find that $\alpha + \beta$ must be 90 degrees, which is Thales’s theorem.

One last thing: this picture also leads to a formula for the area of the right triangle in terms of the length of the hypotenuse. Let d be this length, i.e., the diameter extending from A to C. Look at triangle BOC again. The sum of its angles must also be 180 degrees, and since we know from above that $2 \alpha + 2 \beta = 180$, we find that the missing angle, between segments OB and OC, must be $2 \alpha$. This means the height of our triangle is $\sin (2 \alpha)$ times the radius of the circle, or $\sin (2 \alpha) \cdot d/2$. Since the area of a triangle is half the base times the height, we find

${\rm{Area}} = \frac14 \cdot \sin(2 \alpha) \cdot d^2.$

This last formula is actually connected in a very interesting way to the Pythagorean theorem. But that’s another blog post.