# Pi Day is approaching: What are some interesting math questions whose answer is exactly $\pi$?

In anticipation of Pi Day, which is (of course) March 14, I would like to ask:

What are some interesting math questions whose answer is exactly $$\pi$$?

The questions can be for any age group.

Of course, any math question can be tweaked so that its answer is $$\pi$$. For example, "What is a closed form for $$4\sum\limits_{k=1}^\infty \frac{(-1)^{k-1}}{2k-1}$$ ?", where the $$4$$ is there just to force the answer to be $$\pi$$. But I'm asking for natural, untweaked questions whose answer is exactly $$\pi$$.

Here are some examples.

1. $$n$$ identical "sunrays" of edge length $$1$$ emanate from a circle of radius $$1$$, as shown here with example $$n=8$$. What is $$\lim\limits_{n\to\infty}(\text{total area of rays})$$ ?

I suppose this is not entirely trivial - unless you see the intuitive solution:

Flip the sunrays inside the circle.

1. Evaluate $$\int_{-\infty}^\infty \frac{\sin{x}}{x}dx$$. Here are some solutions.

2. The famous colliding blocks (or balls) question: "Let the mass of two balls be $$M$$ and $$m$$ respectively. Assume that $$M=16 \times 100^n m$$. Now, we will roll the ball with mass $$M$$ towards the lighter ball which is near a wall. How many times do the balls touch each other before the larger ball changes direction? (The large ball hits the small ball which bounces off the wall.)"

• Are you sure the sun-ray example works? To my perception, truly flipping the rays should leave them with indented sides by the circle, not the circle itself... ? Mar 10 at 22:14
• @paulgarrett As the number of rays approaches infinity, the total area of the indentions approaches zero.
– Dan
Mar 10 at 22:17
• Oop, oh-so-true. :) Very good. :) Mar 10 at 22:17
• $\pi$ has taken the larger share of the pie. We need to work $e$ into the calendar, so that hopefully someone will find one simple presentation of it. How about an $e$ moment, to be celebrated at 2 pm, July 18th. Too bad schools are not open in July. Mar 11 at 22:50
• @Maesumi UK schools are but of course the date 7/18 doesn't exist outside North America. (And neither does 3/14, or even 31/4). Mar 12 at 10:46

Regarding your second example. Not only $$\int_{-\infty}^\infty \frac{\sin x}{x} dx = \pi$$, but also $$\int_{-\infty}^\infty \frac{\sin x}{x} \frac{\sin (x/3)}{x/3} dx = \pi$$ $$\int_{-\infty}^\infty \frac{\sin x}{x} \frac{\sin (x/3)}{x/3} \frac{\sin (x/5)}{x/5}dx = \pi$$ $$\int_{-\infty}^\infty \frac{\sin x}{x} \frac{\sin (x/3)}{x/3} \frac{\sin (x/5)}{x/5} \frac{\sin (x/7)}{x/7}dx = \pi$$ $$\dots$$ $$\int_{-\infty}^\infty \frac{\sin x}{x} \frac{\sin (x/3)}{x/3} \frac{\sin (x/5)}{x/5} \frac{\sin (x/7)}{x/7} \dots \frac{\sin (x/13)}{x/13} dx = \pi .$$ And then the pattern breaks and we have $$\int_{-\infty}^\infty \frac{\sin x}{x} \frac{\sin (x/3)}{x/3} \frac{\sin (x/5)}{x/5} \frac{\sin (x/7)}{x/7} \dots \frac{\sin (x/15)}{x/15} dx < \pi$$ specifically

$$\int_{-\infty}^\infty \frac{\sin x}{x} \frac{\sin (x/3)}{x/3} \frac{\sin (x/5)}{x/5} \frac{\sin (x/7)}{x/7} \dots \frac{\sin (x/15)}{x/15} dx \approx \pi - 4.62\times 10^{-11}$$ For the explanation, check the Borwein integral on Wikipedia and this video.

• Well, that explains why I was unable to prove the general case by induction.
– Dan
Mar 11 at 1:23
• @Dan It ultimately boils down to when the sum of the fractions exceeds one, indeed picking a series that never exceeds one we would always have the result to be $\pi$. Mar 11 at 5:28

https://en.wikipedia.org/wiki/Buffon%27s_needle_problem 17 matches thrown randomly. Spacing between lines is 1 match length. 11 cross a line. $$\frac{2 \cdot 17}{11} \approx 3.1 \approx \pi$$.

• Years ago, I worked with several colleagues as one of a few graduate student advisors on a batch of undergraduate research projects. One of the groups was interested in generalizing this result to more spaces. They wrote a nice little paper on the topic: arxiv.org/abs/2009.06755 . Mar 11 at 16:01

"What are some interesting math questions whose answer is exactly $$\pi$$?"

The area of a unit circle--It is exactly $$\pi$$.

The proof is very neat and visual and can be understood by any audience(which is is what separates it from other examples and makes it so important to know.): • That is, if you accept (or define) that $\pi$ is half the ratio of a circle's circumference to its radius. Mar 12 at 0:09
• @TorstenSchoeneberg Yes that is the usual definition. Mar 12 at 5:50
• But this definition is not understood by every audience (which is what Torsten most likely meant to say with his comment). Mar 12 at 13:14
• @Jasper I think any audience with a middle school math education, should know it. Also, given the level of complexity of the other answers, don’t you think that this is a bit of a strawman? Mar 12 at 13:20
• @TorstenSchoeneberg I’d address the same comment to you. Mar 12 at 13:21

The answer is about $$61$$%. Here $$\pi$$ makes its appearance in a surprising, non-circular context (surprising to me):

$$\frac{1}{\zeta(2)} = \frac{6}{\pi^2} \approx 0.6079 \;.$$ And $$\frac{1}{\zeta(d)}$$ in dimension $$d$$: Almost all the lattice points are visible as $$d$$ grows (because $$\zeta(d) \to 1$$).

• I've been pointed to the Basel problem video, which shows an underlying connection to circles: YouTube. Mar 14 at 13:35

On Youtube, there's a video about two squares, one of $$1$$kg. and one of $$10^n$$kg., who are bouncing between themselves and the Y-axis. apparently the amount of bounces equals the decimals of $$\pi$$:

Not an answer, but more of a chuckle. The Indiana Bill also gives $$\pi$$.
More seriously, it all depends on your audience. There are many choices. Some of your students might like the the history of $$\pi$$ as well.