Total surface of a pentagonal pyramid = (5/2)ab + (5/2)bs
Base surface = (5/2)ab
Sides surface = (5/2)bs
Where b is the size of the side, s is the slant height and a is the distance from the center the pentagonal base to the center of one of its sides
Since we have the slant height and the size of each sides, but we don't have "a", we have to calcualte it
The distance "a" can be calculated by:
tan 36° = (b/2)/a ==> a = (b/2)/tan 36° = (4/2)/0.726542528 = 2/0.726542528 = 2.752763841
a = 2.752763841
Now we can calcualte the surface:
Surface = (5/2)ab + (5/2)bs = (5/2) * 2.75 * 4 + (5/2) * 4 * 12 = 27.5 + 120 = 147.5 square inches
Answer:
Total surface = 147.5
Base surface = 27.5
Sides surface = 120