- [Narrator] An ellipse is a circle's more eccentric and interesting companion. If we smoothly squish or pull, maybe not the most technical terms, but you know, a circle, we can create and ellipse. Even though not every oval shape is an ellipse, ellipses are pretty common. For example, the top of a coffee mug is a circle, but seen at an angle, it looks like an ellipse. And the shadow of a disc is often an ellipse. Famously, a circle's circumference is 2 pi r, and presumably, an ellipse has a formula for its circumference too. But weirdly, I don't remember that formula. I'm not sure I learned it in school. Maybe it's like a circle's? So the distance from the center of a circle to its edge is the radius. We usually draw the radii that we use for the formulas straight across, but anywhere works, including at a right angle, and we can rewrite the circumference formula to make this more obvious. On the other hand, the points that make up an ellipse are different distances away from the center so there's no one radius. We often label the longest distance "a" and the shortest distance "b". So using the circumference of a circle as a model, we might imagine that the circumference of an ellipse is pi times a plus b. Is it right? Let's check. I've cut this ellipse out of cardboard. It's long axis a is 5 inches and its short axis b is 1 inch. What does the formula predict? Plugging in some numbers, we get that it should be almost 19 inches long. The circumference is actually really close to 21 inches, something like 10% off, I think we can do better. I hear you screaming - it's a new feature in online video - that we should have used the average of a and b for the radius, but it only takes a little simplifying to see that that's the same formula as we've used before, and it still doesn't work. So let's try something different. There's another kind of average, though. Instead of the arithmetic mean, which uses addition and subtraction, in the geometric mean, you multiply and take the nth root, and with the name that has the word geometry in it, it seems like this might be the way to go. But it gets me thinking, both formulas aren't a disaster. They give the right answer for a circle, and a circle is an ellipse, surely the most boring kind of ellipse since a and b are the same, but an ellipse nonetheless, and we can use that as the standard to try to examine other possibilities. For example, there's another sort of average that kind of combines both the arithmetic and the geometric averages. Let's give it a try. Let's check to see if it gives the right answer for a circle, where a and b are the same value. We'd expect the usual circumference equals 2 pi r, but we get an extra square root of 2. And so, this formula won't work. But we can fix it! If we divide stuff inside the radical by 2, then we get the formula for a circle, and it doesn't guarantee that this is a good formula, but it's a good sign. So though kind of crazy, it might be right. In fact, internet searches for the formula for the perimeter or circumference of an ellipse give this answer dozens of times, but is it right? All there is to do is to try some numbers, and no. It's still off. As a reminder, the length is just about 21 inches, and this new formula gives a value of 1.65 inches bigger than that. So it's still no good. And if you can't trust Chegg to have the right answer, who can you trust? Ready to give up? No, of course not, it just turns out that the formula for the circumference of an ellipse is perhaps just not quite what you were expecting. To understand, let's look back at the formula for the circumference of a circle. You might think that we know the value of pi by measuring a circle's circumference and dividing by twice the radius. That works okay. If I'm careful, I can get 3.1 or maybe 3.14, but it won't do me much better than that. You may have heard about contests where people try to recite hundreds of digits of pi. You can't get that many digits by measuring a real circle and using a real tape measure. A physical circle like this disc wouldn't actually be close enough to a perfect circle, and the tape measure wouldn't be accurate enough, and my skill at measuring wouldn't be good enough. Yet we know pi to billions of places. In 1882, it was shown that pi can't be found by any normal algebraic equation. It can't be the solution of any finite polynomial with rational coefficients. Instead, we have to use another way to find the value of pi. In the 15th century, the Indian mathematician and astronomer, Nilakantha Somayaji, discovered this method, and there are others, but I like this one. Start with 3, then add 4 divided by 2 times 3 times 4. You'll get 3.166 and so on. Subtract 4 divided by 4 times 5 times 6, add 4 divided by 6 times 7 times 8, subtract 4 divided by 8 times 9 times 10, then add 4 divided by 10 times 11 times 12, and keep on going as long as you like, to be as precise as you like. I won't explain how this worked right at this time. I'll leave that for another time. Even with all of that amazing precision, and as you can see from the above equations, pi alone cannot accurately find the circumference of an ellipse. Pi just isn't quite the right value. It turns out that pi needs friends, friends like pi itself, made up of an infinite series of terms, friends that are at least as complicated as pi. No special letters are given to these friends because pi actually needs lots of friends to find the circumference of lots of different ellipses. Pi needs an infinity of friends. Remember that a and b can tell us the size and shape of an ellipse. All those different infinite varieties of ellipses, each has its own number that can team up with pi to find its circumference. You might be wondering, "How did we miss this?" Well, we were blinded by how well pi times a plus b works to find the area of a circle. We don't notice the friend that is helping pi because the friend for a circle is 1. When we multiply pi times 1, it stays pi since any number times 1 doesn't change. We don't even know that it's there, but all of the rest of the helper numbers are much more complicated. Like pi, they are computed by adding up a series of terms. The more terms you use, the more precise you can be. For our ellipse, where a was 5 and b was 1, we get the helper number to be 1.11432. Let's use our helper to find, finally, the perimeter of our ellipse. Pi times 1.11432 times 5 plus 1 is 21.004. Pi times 1.11432 times 5 plus 1 is 21.004. Yes! And looking back at our physical ellipse, its circumference really is about 21 inches, but don't get complacent. Pi's friend, 1.11432, doesn't work for every ellipse, only ones that have the same ratio of a to b as our ellipse, and there are an infinite number of other possible shapes. Pi will need lots of friends. To find the friend for each ellipse, we can use this terrible-looking monstrosity, but it isn't too bad since this part for any particular ellipse stays the same. With careful computation, you can finally find pi's friend, who when multiplied to pi can find the circumference of an ellipse as precisely as you'd want. It's amazing to discover what we can do with a little help from our friends.
When Pi isn't Enough
Published: March 8, 2021
Total Running Time: 00:08:20
You can use pi to find the circumference of a circle, but it won’t help you find the circumference of an ellipse. Find out about pi's kryptonite, and how it takes a team of numbers to save our hero.