SOLUTION: The sum of the interior angles of a regular polygon is 1,260. find the area of the polygon if its perimeter is 45 centimeters.

Algebra ->  Polygons -> SOLUTION: The sum of the interior angles of a regular polygon is 1,260. find the area of the polygon if its perimeter is 45 centimeters.       Log On


   

Question 815167: The sum of the interior angles of a regular polygon is 1,260. find the area of the polygon if its perimeter is 45 centimeters.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The interior angles of a polygon with N sides add up to
%28N-2%29%2A180%5Eo .
In this case,
%28N-2%29%2A180=1260 --> N-2=1260%2F180 --> N-2=7 --> N=9 .
So we are dealing with a polygon with 9 sides.
One of the formulas to calculate the area of a polygon is
Area+=+perimeter%2Aapothem%2F2=N%2Aside%2Aapothem%2F2 ,
where apothem is the segment or the distance from the center of the polygon to the center of one of its sides.
That "formula" comes from adding up the area of the N isosceles triangles formed by connecting the vertices to the center of the polygon.
The apothem (meaning the distance, the length of the apothem segment) can be calculated as
%28side%2F2%29%2Ftan%28180%5Eo%2FN%29 .
In turn, the length of a side can be calculated from the perimeter and N as
side=perimeter%2FN
The whole calculation can be done step by step, or put together as one fancy formula.
45%2F9=5 is the side (in cm)
180%2F9=20
tan%2820%5Eo%29=0.36397
5%2F2%2F0.36397=6.8687 is the apothem (in cm), and the area is
45%2A6.8687%2F2=154.6 square centimeters.