Answer:
To find the total surface area of a square pyramid, we need to find the sum of the areas of its base and its four triangular faces.
In this case, the base is a square with side length 11.5 inches. Therefore, its area is:
Area of base = 11.5^2 = 132.25 square inches
Each triangular face has a base length equal to the side length of the square base, which is 11.5 inches. To find the height of each triangular face, we can use the Pythagorean theorem. Since the pyramid is a square pyramid, the height of each triangular face is also the slant height of the pyramid.
The slant height of the pyramid can be found using the Pythagorean theorem:
a^2 + b^2 = c^2
where a = b = 11.5/2 = 5.75 (half the length of a diagonal of the square base) and c is the slant height. Solving for c, we get:
c = sqrt(a^2 + b^2) = sqrt(2*(5.75)^2) = 8.121 inches (rounded to 3 decimal places)
The area of each triangular face can be found using the formula:
Area of triangle = (1/2) * base * height
where the base is 11.5 inches and the height is 8.121 inches.
Area of each triangular face = (1/2) * 11.5 * 8.121 = 46.876 square inches (rounded to 3 decimal places)
So, the total surface area of the square pyramid is:
Total surface area = Area of base + 4 * Area of each triangular face
Total surface area = 132.25 + 4 * 46.876 = 330.124 square inches (rounded to 3 decimal places)
Therefore, the total surface area of the square pyramid is approximately 330.124 square inches.