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' Answer by htmentor(1343) (Show Source):
You can put this solution on YOUR website!
The pentagon can be broken up into 5 triangles.
To find the area, we need to find the area of one of the triangles and multiply by 5.
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.
It is also clear that the apothem is the height of one of these triangles.
The angles whose vertex is the center of the circle must sum to 360 degrees, which means each one is 72 degrees.
The measure of the half-angle is 36 degrees
From the figure, we see that tan(36) = BG/KG -> BG = KG*tan(36)
The area of the shaded triangle is BG*KG, and the area of the pentagon is 5*BG*KG
Substituting, we get Area = 5*KG*KG*tan(36) = 5*35^2*tan(36) = 4450.1
