A Tenth Of A Pizza

On Twitter, @Trianglemanscd posed a pertinent problem:

Stand back everyone! I have compasses and a straight-edge and I’m not afraid to use them; the Geogebra demonstration below shows one way to do it, eschewing things like ‘strings’ and ‘protractors’ in favour of proper geometry.

The nugget to this approach is that $\cos(36^o) = \frac{\sqrt{5}-1}{4}$. That’s closely related to the golden ratio $\phi$ – in fact, it’s $\frac{\phi}{2}$.

So all we need to do is construct a distance of $\frac{\phi}{2}$ (which is the first five steps) and then a right-angled triangle with a hypotenuse of 2.

The angle at the centre is $\frac{\pi}{5}$, or 36 of your silly degrees - a tenth of a pizza.


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.


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Where do you teach?

I teach in my home in Abbotsbury Road, Weymouth.

It's a 15-minute walk from Weymouth station, and it's on bus routes 3, 8 and X53. On-road parking is available nearby.

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