Find the area of the regular polygon.
pentagon with a radius of 4 ft

In this problem we need to find the apothem and the length of the side before we can find the area of the entire polygon. Each central angle for a regular pentagon is 360∘5=72∘. So, half of that, to make a right triangle with the apothem, is 36∘. We need to use sine and cosine. 
sin36∘4sin36∘8sin36∘n=.5n4=12n=n≈4.7 cos36∘=a44cos36∘=aa≈3.24
Using these two pieces of information, we can now find the area. 
A=12(3.24)(5)(4.7)≈38.07 units^2.