Answer:
about 840.4 ft²
Explanation:
You want the surface area of a regular pentagon with a side length of 16 ft and a slant height of 10 ft.
Area
The area of the 5-sided base is ...
A = 1/4(√(5(5 +2√5))s² . . . . for side length s
The area of the 5 triangular faces is ...
A = 5×(1/2)sh
Using s = 16 ft and h = 10 ft, we find the total surface area to be ...
SA = 1/4(√(5(5 +2√5))(16 ft)² +5/2(16 ft)(10 ft) ≈ 840.4 ft²
The surface area of this pentagonal pyramid is about 840.4 square feet.
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Additional comment
The problem statement says the figure has a hexagonal base. We only count five sides, so have worked the problem as though it has a pentagonal base.
Either way, the figure cannot exist. The lateral area of such a figure must exceed the base area. In this figure, the base area is about 10% greater than the lateral area, meaning the points of the triangular faces cannot meet to form a pyramid.
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