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Question 1164596: the number of diagonals of a regular polygon is 35. find the area if it's apothem is 15
Answer by htmentor(1343)   (Show Source):
You can put this solution on YOUR website! The number of diagonals of an n-sided polygon is: (n/2)(n-3)
Thus, n/2(n-3) = 35 -> (n-10)(n+7) = 0 -> n = 10 (take the positive solution)
The polygon is made up of 10 triangles, with altitude equal to the apothem
The interior angles of a decagon are 144 deg.
The base of each triangle, b = 2a/tan(144/2), where a = the apothem
So the area of each triangle = a/tan(72)*a = 15^2/tan(72)
Thus the entire area is 10*15^2/tan(72) = 731.07
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