document.write( "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. \n" ); document.write( "

Algebra.Com's Answer #491516 by KMST(5328) $\"\"$ $\"About$

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The interior angles of a polygon with N sides add up to
\n" ); document.write( " $\"%28N-2%29%2A180%5Eo\"$ .
\n" ); document.write( "In this case,
\n" ); document.write( " $\"%28N-2%29%2A180=1260\"$ --> $\"N-2=1260%2F180\"$ --> $\"N-2=7\"$ --> $\"N=9\"$ .
\n" ); document.write( "So we are dealing with a polygon with 9 sides.
\n" ); document.write( "One of the formulas to calculate the area of a polygon is
\n" ); document.write( " $\"Area+=+perimeter%2Aapothem%2F2=N%2Aside%2Aapothem%2F2\"$ ,
\n" ); document.write( "where apothem is the segment or the distance from the center of the polygon to the center of one of its sides.
\n" ); document.write( "That \"formula\" comes from adding up the area of the N isosceles triangles formed by connecting the vertices to the center of the polygon.
\n" ); document.write( "The apothem (meaning the distance, the length of the apothem segment) can be calculated as
\n" ); document.write( " $\"%28side%2F2%29%2Ftan%28180%5Eo%2FN%29\"$ .
\n" ); document.write( "In turn, the length of a side can be calculated from the perimeter and N as
\n" ); document.write( " $\"side=perimeter%2FN\"$
\n" ); document.write( "The whole calculation can be done step by step, or put together as one fancy formula.
\n" ); document.write( " $\"45%2F9=5\"$ is the side (in cm)
\n" ); document.write( " $\"180%2F9=20\"$
\n" ); document.write( " $\"tan%2820%5Eo%29=0.36397\"$
\n" ); document.write( " $\"5%2F2%2F0.36397=6.8687\"$ is the apothem (in cm), and the area is
\n" ); document.write( " $\"45%2A6.8687%2F2=154.6\"$ square centimeters.
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