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Re: If the sum of the interior angles of a regular polygon... [#permalink]
05 Oct 2010, 05:56
If you don't know the formula you can get to the answer this way. A little more involved but probably still doable in the crunch:
Start with the second to highest number in the answers (i.e. 10) so you can go higher or lower depending on what you find.
Draw or visualize a decagon, and run the five diagonals that go through the center, so that you end up with 10 identical 'wedges' or 'pizza slices' which will be isosceles triangles.
All 10 triangles will have the center, where all diagonals cross, in common. You know the sum of all those angles is 360, so since you have 10 of them, that angle is 360/10 = 36 for each triangle. Now you can see that for each slice, the sum of the other triangles must be 180 - 36 = 144. Therefore, since you have 10 triangles, the total sum of the angles that make the decagon is 144 *10 = 1440. Voila.
If this had been too low, you know the polygon would have to have more sides, if it had been too big, you would need less. so you could work that way.
Still, formula is MUCH simpler so this tells me I need to brush up!!