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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 $n=8$.

enter image description here

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.)"

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  • $\begingroup$ 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... ? $\endgroup$
    – paul garrett
    Mar 10 at 22:14
  • $\begingroup$ @paulgarrett As the number of rays approaches infinity, the total area of the indentions approaches zero. $\endgroup$
    – Dan
    Mar 10 at 22:17
  • $\begingroup$ Oop, oh-so-true. :) Very good. :) $\endgroup$
    – paul garrett
    Mar 10 at 22:17
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    $\begingroup$ $\pi$ 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. $\endgroup$
    – Maesumi
    Mar 11 at 22:50
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    $\begingroup$ @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). $\endgroup$
    – Andrew Leach
    Mar 12 at 10:46

6 Answers 6

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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.

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    $\begingroup$ Well, that explains why I was unable to prove the general case by induction. $\endgroup$
    – Dan
    Mar 11 at 1:23
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    $\begingroup$ @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 $\pi$. $\endgroup$
    – Vivaan Daga
    Mar 11 at 5:28
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https://en.wikipedia.org/wiki/Buffon%27s_needle_problem

enter image description here

17 matches thrown randomly. Spacing between lines is 1 match length. 11 cross a line. $\frac{2 \cdot 17}{11} \approx 3.1 \approx \pi$.

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    $\begingroup$ 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 . $\endgroup$
    – Xander Henderson ♦
    Mar 11 at 16:01
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"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.):

enter image description here

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    $\begingroup$ That is, if you accept (or define) that $\pi$ is half the ratio of a circle's circumference to its radius. $\endgroup$
    – Torsten Schoeneberg
    Mar 12 at 0:09
  • $\begingroup$ @TorstenSchoeneberg Yes that is the usual definition. $\endgroup$
    – Vivaan Daga
    Mar 12 at 5:50
  • $\begingroup$ But this definition is not understood by every audience (which is what Torsten most likely meant to say with his comment). $\endgroup$
    – Jasper
    Mar 12 at 13:14
  • $\begingroup$ @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? $\endgroup$
    – Vivaan Daga
    Mar 12 at 13:20
  • $\begingroup$ @TorstenSchoeneberg I’d address the same comment to you. $\endgroup$
    – Vivaan Daga
    Mar 12 at 13:21
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A favorite is: What fraction of the integer lattice can be seen from the origin?.

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

$$\frac{1}{\zeta(2)} = \frac{6}{\pi^2} \approx 0.6079 \;.$$

LatticeVisib8Grid

And $\frac{1}{\zeta(d)}$ in dimension $d$: Almost all the lattice points are visible as $d$ grows (because $\zeta(d) \to 1$).

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  • $\begingroup$ I've been pointed to the Basel problem video, which shows an underlying connection to circles: YouTube. $\endgroup$
    – Joseph O'Rourke
    Mar 14 at 13:35
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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 $\pi$:

Hereby the Youtube URL, uploaded by 3blue1brown.

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    $\begingroup$ This is number 3 in the OP, by the way, but still it's useful to have a Youtube video reference. You may want to add the author and the title, so that your answer is more self-contained. $\endgroup$
    – Michał Miśkiewicz
    Mar 15 at 10:29
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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.

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