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.

[iframe src=”https://www.geogebra.org/material/iframe/id/ahws4xvy/width/1022/height/539/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false” width=”1022px” height=”539px” scrolling=”no” title=”A tenth of a pizza” ]

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.