Here's a tasty puzzle.
Alice would like to share a slice of pie with Bob. Alice doesn't like crust so she decides to cut the pie cross-wise. However, she wants to make sure that she's giving herself and Bob an even amount.
Given the scenario below, how far up the pie edge (r/R) should Alice make her cut?
a = Alice's slice area, b = Bob's slice area
If the angle of the slice is $\theta$, then $a = \dfrac{1}{2}r^2 \sin\theta$ and $b = \dfrac{1}{2}R^2 \theta -a$. For them to be equal $$r = \sqrt{\frac{\theta}{2 \sin\theta}}\; R$$ Note that eight slices means $\theta = \dfrac{\pi}{4}$.