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.

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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)   (Show Source): You can put this solution on YOUR website!
The interior angles of a polygon with N sides add up to
.
In this case,
--> --> --> .
So we are dealing with a polygon with 9 sides.
One of the formulas to calculate the area of a polygon is
,
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
.
In turn, the length of a side can be calculated from the perimeter and N as

The whole calculation can be done step by step, or put together as one fancy formula.
is the side (in cm)


is the apothem (in cm), and the area is
square centimeters.
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