How to approximate $0.714286$ as a fraction of $\pi$?

Question

I'm doing an exercise that tells me that the answer must be a multiple of pi, like $12\pi$ or $\dfrac23\pi$.

I need to approximate $0.714286$ as a fraction of $\pi$.

How do I achieve this?

Answer 1

You may compute the continued fraction for $\frac{\pi}{C}$ where $C$ is your constant: $$ \frac{\pi}{C}=[4;2,1,1,21,1,35,\ldots] $$ and by expanding $[4,2,1,1]=\frac{22}{5}$ we get: $$ C = 0.714286 \approx \frac{5\pi}{22}.$$ A better approximation is given by expanding $[4,2,1,1,21,1]=\frac{497}{113}$, so $$ C = 0.714286 \approx \frac{113\pi}{497}.$$