Angles in Stars
A fascinating property of the -pointed star, , is that its sum of angles at the "points" does not depend on the polygon chosen to construct it. It only depends on the and chosen.
What is the the sum of angles at the "points" of the 201-pointed star, , in degrees?
Note: denotes a -pointed star formed by joining every vertex on a convex -gon.
The answer is 33660.
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1 solution
The formula for the sum of the angles is 1 8 0 p − 3 6 0 q .
The reasoning behind it:
Every vertex contributes 1 8 0 ∘ minus the external angle at that vertex.
The sum of all external angles is equal to the number of times the self-intersecting star polynomial goes around the outside.
This in turn is equal to q , because by taking, in our case, every 7th vertex, we have to go around another six times to get all those skipped, making a total of 7 trips around.