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Bot GPT 3.5 To calculate the surface area of a regular hexagonal pyramid, you need to find the areas of the six triangular faces and the area of the hexagonal base.
The area of each triangular face can be calculated using the formula:
A = 1/2 * base * height
Since the base of the pyramid is a regular hexagon, we can split it into 6 equilateral triangles with side lengths equal to the base length. The height of each triangle can be found using the Pythagorean theorem:
height = sqrt(slant height^2 - (base/2)^2)
height = sqrt(9.8^2 - (8/2)^2)
height = sqrt(95.04 - 16)
height = sqrt(79.04)
height ≈ 8.89 in
Now we can calculate the area of each triangular face:
A = 1/2 * 8 * 8.89
A ≈ 35.56 sq in
The total area of the six triangular faces is:
6 * 35.56 = 213.36 sq in
To calculate the area of the hexagonal base, we use the formula for the area of a regular hexagon:
Area = 3 * sqrt(3) * (side length)^2
Area = 3 * sqrt(3) * 8^2
Area = 3 * sqrt(3) * 64
Area = 3 * 8 * sqrt(3)
Area ≈ 41.57 sq in
Finally, the total surface area of the regular hexagonal pyramid is the sum of the areas of the triangular faces and the base:
Total surface area = 213.36 + 41.57
Total surface area ≈ 254.93 sq in
Therefore, the surface area of the regular hexagonal pyramid is approximately 254.93 square inches.