If there are n points on the circumference of a "cake" and each pair of these points is joined by a line or "cut" Let Wn be the number of regions. In the attached picture W4 = 8 (points ABCD). If we add point E and join it to every other point. Consider how many regions the segment AED is split into and then triangles ABC and ABD (for EB) and ACD and BCD (for EC)and generalizing show that: Wn+1 = Wn + n + n(n-1)(n-2)/6.
Cutting a Circle into Triangles is investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.