Question

Find the area of a regular pentagon whose apothem is 25.1 mm and perimeter is 182 mm.

Answer 1

Area = (182 mm)(25.1 mm) / 2 = 2284.1 mm^2
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Proof for Area of a Regular Polygon:
It's not that hard to come up with by yourself.

Imagine you had any regular polygon. Now radially divide it into triangles (draw lines from all the vertices to the center). There is a triangle for every side of the polygon. The apothem is the line from the center to the center of a side. Notice that this is just the height of a triangle. Obviously, the area of the regular polygon would be the added areas of all the triangles. The area of a triangle is (1/2)bh which in this case is (1/2)SA where s is the side length and A is the apothem. The number of triangles equals the number of sides so the area of the regular polygon is N(1/2)SA where N is the number of sides. For a regular polygon, the perimeter, P = NS. So, the area for a regular polygon is AP/