# A proof that π < 22/7

Note: this proof has a fatal error.

There’s a reasonably well-known proof that $22/7 > \pi$, which can be written in one line: $0 < \int_0^1 {x^4 (1-x)^4 \over 1+x^2} dx = {22 \over 7} - \pi$ But I’ve always found this one unsatisfying because what does that integral have to do with $\pi$ anyway? As it turns out, $\pi$ enters through the integral $\int_0^1 {1 \over (1+x^2)} = \pi/4$. But let’s say I’m a purist and think that $\pi$ is about circles. Can I do better? (Of course I can. If I couldn’t I wouldn’t be writing this post.) Start by observing that $(21/11)^3 > 7$, which can be shown by explicit computation: $11^3 = 1331$ and $21^3 / 7 = 21^2 \times 3 = 1323$.

Edited to add, November 30: of course I just showed here that $(21/11)^3 < 7$. This is what happens when you do arithmetic in your head…

Now, the sine function is given by the alternating series $sin(x) = x - {x^3 \over 3!} + {x^5 \over 5!} - {x^7 \over 7!} + \cdots$ and in particular $\sin(x) > x - x^3/6$ by the alternating series test. Applying this with $x = 11/21$ gives $\sin(11/21) > 11/21 - (11/21)^3/6 > 11/21 - (1/7)/6 = 1/2$.

Taking the inverse sine of both sides, $11/21 > \sin^{-1} 1/2$.

Finally, we have $\pi = 6 \sin^{-1} (1/2)$. This is a geometric fact that goes back to Euclid’s construction of the hexagon. So $\pi < 22/7$.

On related notes:

•  Noam Elkies, Why is π2 so close to 10?
• Alejandro Morales, Igor Pak, and Greta Panova, Why is π < 2φ?  (This is not a particularly good approximation, but it actually admits a combinatorial proof in terms of Fibonacci and Euler numbers.)

## 2 thoughts on “A proof that π < 22/7”

1. zl84841g says:

Nice little proof. Their must be a problem with your TeX code. The integral is not right. Is there a missing ‘\’ before your over?

\int_0^1 \frac{x^4(1-x)^4}{1+x^2} \, dx = \frac{22}{7} – \pi

looks to be correct code, stolen from https://en.wikipedia.org/wiki/Proof_that_22/7_exceeds_π. I hope wordpress does not mangle the TeX I posted.

2. Raghav says:

The proof is actually incorrect, (21/11)^3 is not greater than 7.