# A Tenth Of A Pizza

On Twitter, @Trianglemanscd posed a pertinent problem:

Making pizzas tonight but cannot recall how to cut them in 10 slices.

— Christopher Danielson (@Trianglemancsd) May 31, 2019

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.

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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.