Question

Find the apothem of a regular polygon with an area of 625 m2 and a perimeter of 100 m

Answer 1

The area of a polygon is a collection of triangles whose base is a portion of the perimeter, and the height is the apothem. From the attached picture we get:\[Area_{Total}=(Area_{Triangle})*(NumberOfSides)\]
\[Area_{Total}=( \frac{1}{2}b*h )*(NumberOfSides)\]
\[Area_{Total}=( \frac{1}{2}\frac{Perimeter}{NumberOfSides}*h )*(NumberOfSides)\]
\[625= \frac{1}{2} {Perimeter} *h )\]
\[h=\frac{2*625}{Perimeter}\]
\[h=\frac{2*625}{100}\]
which is independent from the number of sides the polygon has:
\[apothem = 2*\frac{Area}{Perimeter}\]