Calculate the surface area of the square pyramid that has a base length of 12 and a slant height of 18. Show your work.(2 points).
To calculate the surface area of a square pyramid, we need to find the area of the base and the area of the four triangular faces.
1. Area of the base:
Since the base is a square with a side length of 12, we can find the area by squaring the side length:
Area of base = side length squared
Area of base = 12^2
Area of base = 144
2. Area of each triangular face:
Since the slant height of the pyramid is given as 18, and the base side length is 12, we can use the Pythagorean theorem to calculate the height of each triangular face:
height = sqrt(18^2 - (12/2)^2)
height = sqrt(324 - 36)
height = sqrt(288)
height = 12√2
Now, we can find the area of each triangular face using the base side length of 12 and height of 12√2:
Area of each triangular face = 0.5 * base length * height
Area of each triangular face = 0.5 * 12 * 12√2
Area of each triangular face = 72√2
3. Total surface area:
Total surface area = area of base + 4 * area of each triangular face
Total surface area = 144 + 4 * 72√2
Total surface area = 144 + 288√2
Therefore, the surface area of the square pyramid with a base length of 12 and a slant height of 18 is 144 + 288√2 square units.