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.

sin 36∘ 4sin 36∘ 8sin 36∘ n =.5n 4 = 12 n = n ≈4.7 cos 36∘ = a4 4cos 36∘ = a a≈ 3.24

Using these two pieces of information, we can now find the area.

A = 12 (3.24) (5) (4.7) ≈38.07 unit s^2.