The surface area of the entire pentagonal pyramid roof is; 399.2975 ft²
What is the surface area of the Pyramid?
Since the slant height is given, the surface area of the triangular faces is easy to find, with each triangular face having an area of (1/2)bh, or (1/2) (7 * 9 = 63. Since there are 5 triangular faces, the surface area of all the triangular faces together is 5(63) = 315 ft²
The area of the base of the pentagon must be found from the edge length.
Since the apothem (distance from the midpoint of the edge to the center of the pentagon) is not given, it must be calculated. If you divide the pentagon into 5 equal triangles from the center, the top angle of one of these triangles is 360/5 = 72°.
Dividing this isosceles triangle in half gives two right triangles. The angle at the top is 1/2 of 72° = 36°. The tangent of this angle can be expressed as side opposite divided by side adjacent, or half the edge (7/2) divided by the apothem (a). Thus;
tan 36° = side opposite/side adjacent = 3.5/a
Thus;
a = 3.5/tan 36
a = 4.817 ft
The area of each of the 5 triangles making up the pentagon base is;
(1/2)ba = (1/2)(7)(4.817) = 16.8595 ft²
The area of the pentagon base is therefore 5(16.8595) = 84.2975 ft²
The surface area of the entire pentagonal pyramid is therefore;
315 + 84.2975 = 399.2975 ft²
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