To find the surface area of a square pyramid, you need to find the area of the base and the area of the four triangular faces.
The area of the base of a square pyramid is equal to the side length squared. In this case, the base length is 3 ft, so the area of the base is 3 ft * 3 ft = 9 ft^2.
The area of each triangular face can be found using the formula for the area of a triangle: 1/2 * base * height. In this case, the base is 3 ft (the same as the side length of the base), and the height can be found using the Pythagorean theorem: height^2 + base/2^2 = slant height^2. Plugging in the given slant height of 8 ft, we have height^2 + (3/2)^2 = 8^2. Simplifying, height^2 + 9/4 = 64. Subtracting 9/4 from both sides, we have height^2 = 64 - 9/4 = 256/4 - 9/4 = 247/4. Taking the square root of both sides, we have height = sqrt(247)/2 ft.
Now, we can find the area of each triangular face using the formula: 1/2 * base * height. Plugging in the base length of 3 ft and the height of sqrt(247)/2 ft, we have 1/2 * 3 ft * sqrt(247)/2 ft = 3sqrt(247)/4 ft^2.
Since there are four triangular faces, the total area of the triangular faces is 4 * 3sqrt(247)/4 ft^2 = 3sqrt(247) ft^2.
Finally, to find the total surface area, we add the area of the base to the area of the triangular faces: 9 ft^2 + 3sqrt(247) ft^2.
So the correct answer is 9 ft^2 + 3sqrt(247) ft^2.
Find the surface area of a square pyramid with a base length of 3 ft and a slant height of 8 ft.
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1 point
63 ft^2
57 ft^2
48 ft^2
72 ft^2
1 answer