Answer
$A=5r^2cos36^{\circ}sin36^{\circ}$
Work Step by Step
When we break a regular pentagon into 10 separate right triangles, each triangle has an angle of 36 degrees in it. Using this angle and the fact that the radius is r, we find:
$sin36^{\circ}=\frac{s/2}{r} \\ s/2 = rsin36^{\circ} \\ s = 2rsin36^{\circ}$
Since there are 5 sides, we find:
$P = 5s = 10rsin36^{\circ}$
The area is equal to the 1/2 times perimeter times the apothem. Thus, we find the apothem:
$cos36^{\circ}= \frac{a}{r} \\ a = rcos36^{\circ}$
Thus, we obtain:
$A = \frac{1}{2} rcos36^{\circ}10rsin36^{\circ} =5r^2cos36^{\circ}sin36^{\circ}$