Extensions

Consider other type regions such as the one below such that regions 1, 2, and 3 are equal.

Or how about this one? Again regions 1, 2, and 3 are equal.

And here is one in which the three regions are cut from the circle by shifted sine waves! What sine waves could be used?

Can one consider three sections such that the OUTER perimeter (a tasty extra thick icing perhaps) is the same for all three sections.

Can one consider three sections such that the TOTAL perimeter (a tasty extra thick icing perhaps) is the same for all three sections.

Of course dividing the circle into three equal cuts (120 degrees each) through the center gives each participant equal area AND equal perimeter. Are there other "nice" cuts which do this, e.g., can any of our cuts do this? can they be modified to do this?

One could also consider the problem of cutting the cake into say n (n > 3) equal pieces and prescribing some geometrical constraints.