Friday, September 16, 2011

Generalization of Heron's Formula

I thought if you start off with some kind of general convex region and all you know is the lengths of the segments around the outside and the order they go in, you could the following:
Connect every other vertex going around back to the beginning unless you're on the last vertex, in that case just connect around to the beginning.
Then iterate that process over the last vertices that were connected:
So until you get to just the last 3 vertices every step eliminated at least 2 vertices, so if there started off with N vertices it would take at most (N-3)/2 steps. Anyway if you know the N sides around, this process creates less than or equal to 2/3 but more than 1/2 new lines. And every knew line creates an additional triangle.
  Anyway that point of analyzing that is that there are more knowns necessarily than unknowns in the system as a whole and enough equations from Heron's formula for each triangle that the length of every line can be solved for algebraically, so therefore the total area  can be known from adding them all together. Which is pretty surprising considering that at first thought it seems that possibly all the outside could be adjusted somehow to make a shape with the same perimeter and side lengths but different area, but apparently not.

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