I originally found this question in James Stewart's Calculus, specifically in one of the Problems Plus sections.
The question asks how $3$ people can share a pizza while making just $2$ cuts, instead of the usual $3$. The basic idea is to divide a circle into $3$ equal pieces using $2$ parallel lines that are the same distance from the center, as in the picture below.
I am having trouble getting started on this question, mainly because I have no idea how to find the areas of the $3$ pieces. I can see that the cuts should not be too close to the edges or the center, as this would make middle piece too large or too small respectively. But I cannot see much more than this.
Best Answer
Assuming that the pizza has unit radius, the area of the pizza is $\pi$ and each piece has to have area $\frac{\pi}{3}$. So it is enough to find the height of a circle segment such that its area is one third of the original circle, i.e. to solve
$$ \int_{-\sqrt{h(2-h)}}^{\sqrt{h(2-h)}}\sqrt{1-x^2}\,dx - 2(1-h)\sqrt{h(2-h)}=\frac{\pi}{3}$$ in terms of $h$, that with some substitution boils down to Kepler's equation$^{(*)}$, i.e. to a trascendental equation that, in general, cannot be solved in explicit terms, but is easy to solve numerically by Newton's method. In our case we get $h\approx \color{red}{0.735068}\approx\frac{4492}{6111}$, so the three slices have widths approximately proportional to $3:2:3$.
$(*)$: who guessed that planetary motion and pizza slicing are related?