SOLUTION: The number of diagonals of a regular polygon is 35. Find the area of the polygon if its apothem measures 10 cm.

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Question 630017: The number of diagonals of a regular polygon is 35. Find the area of the polygon if its apothem measures 10 cm.
Answer by AnlytcPhil(1807) About Me  (Show Source):
You can put this solution on YOUR website!
There are n vertices, there are C(n,2) ways to connect 2 of them.
All of these represent diagonals except the n sides, so the number
of diagonals is

     C(n,2) - n = 35 

     %28n%28n-1%29%29%2F2 - n = 35
     
     n(n-1) - 2n = 70

     nē - n - 2n - 70 = 0

     nē - 3n - 70 = 0
   
     (n - 10)(n + 7) = 0

      n = 10,  n = -7, which we discard.

So the regular polygon has 10 sides.

The formula for the area of an n-sided regular polygon is

Area = na%5E2tan%28%22180%B0%22%2Fn%29, where a = apothem and 
n = the number of sides.

Area = 10%2A10%5E2tan%28%22180%B0%22%2F10%29

Area = %2810%2A100%2Atan%28%2218%B0%22%29%29

Area = %281000%29%2Atan%28%2218%B0%22%29%29

Area = 324.9196962 square units of area

Edwin