GoKawiilTau day 2026: Pi square is nearly 10 Posted on June 28, 2026 ; last changed on June 30, 2026
In the US and countries with a similar date format, today is the \(\tau\) day (\(\tau = 2 \pi\)). I still think that \(\tau r\) and \(\frac{\tau r^2}{2}\) are better formulas than \(2\pi r\) and \(\pi r^2\), since they match the \(mv\) and \(\frac{mv^2}{2}\) ones (and many other reasons). But that ship has sailed, so \(\tau\) is relegating to just being the double of \(\pi\).
Well, addition is trivial, but did you know that \(\pi^2 \approx 10\) and \(\pi^2 \approx g\) (where \(g\) is the acceleration due to gravity at sea level on Earth)? How did we get to these coincidences?
For today, let’ just check the first fact. We have \(\pi^2 \approx 9.8696\) which is close to 10 (for certain definitions of 10).
Let’s start with that famous formula where \(\pi^2\) shows up, the Basel problem: what is the value of the sum of the reciprocal of the squares of natural numbers? We know the answer from Euler:
\[ \sum_{n=1}^{\infty}{\frac{1}{n^2}} = \frac{\pi^2}{6} \]
That is
\[ \pi^2 = 6\zeta(2) \]
where \(\zeta\) is the Riemann zeta function. In our case we can do this manipulation
\[ \zeta(2) = \sum_{n=1}^{\infty}{\frac{1}{n^2}} = 1 + \sum_{n=2}^{\infty}{\frac{4}{4n^2}} \]
... continue reading