document.write( "Question 872015: I want to find the area of a regular pentagon and I only have the apothem. Can I do this? The Apothem is 35' \n" ); document.write( "

Algebra.Com's Answer #525899 by htmentor(1343) $\"\"$ $\"About$

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\n" ); document.write( "The pentagon can be broken up into 5 triangles.
\n" ); document.write( "To find the area, we need to find the area of one of the triangles and multiply by 5.
\n" ); document.write( "From the figure, we see that the apothem (in red) is the radius of the in-circle, which touches the pentagon at the midpoints of the sides.
\n" ); document.write( "It is also clear that the apothem is the height of one of these triangles.
\n" ); document.write( "The angles whose vertex is the center of the circle must sum to 360 degrees, which means each one is 72 degrees.
\n" ); document.write( "The measure of the half-angle is 36 degrees
\n" ); document.write( "From the figure, we see that tan(36) = BG/KG -> BG = KG*tan(36)
\n" ); document.write( "The area of the shaded triangle is BG*KG, and the area of the pentagon is 5*BG*KG
\n" ); document.write( "Substituting, we get Area = 5*KG*KG*tan(36) = 5*35^2*tan(36) = 4450.1
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