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:

pentagon

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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