## A circle inscribing a pentagon

This is also from The Irrationals – though I had to ask for assistance over at Stack Exchange to get the right answer (as is so often the case the solution is reasonably obvious when you are presented with it).

Anyway, the question is: given a regular pentagon (of sides with length of 10 units) which is inscribed by a circle, what is the diameter of the circle?

This figure helps illustrate the problem: We are trying to find $2R$ and we know that $x=10$.

If we knew $r$ then we could answer as $R^2 = r^2 + (\frac{x}{2})^2$. We do not know that but we do know that $\sin(\frac{\pi}{5})=\frac{x}{2R}$ and hence, from the $\cos^2 + \sin^2 = 1$ identity: $\sin\frac{\pi}{5} = \frac{10}{2R} = \sqrt{1-\cos^2(\frac{\pi}{5})}$.

From our knowledge of the pentagon with sides of unit length (you’ll have to trust me on this or look it up – it’s too much extra to fit in here) we also know that $\cos(\frac{\pi}{5}) = \frac{\phi}{2} = \frac{1}{4}(1+\sqrt{5})$, where $\phi$ is the golden ratio.

Hence $2R =$ … well, the rest is left as an exercise for the reader 🙂

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## In praise of StackExchange

I have been ill – flu I think – for the last few days and so “blogging has been light”. But I have discovered StackExchange in the meantime.

Of course, I was always aware that StackExchange was there, but I suppose I thought of it as a parallel to “Experts-Exchange” – the awful, pay for a solution, site. But actually, once I overcame my loathing for having to login, I discovered it was a series of communities based on trust and free exchange of information, with acceptable behaviour policed through user activism and a really good “karmic” points system.

I have even been able to answer a couple of questions!

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