Question
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)
Math Short Answer Rubric (2 points)
Points Awarded Criteria
2
The answer is correct. The student made no errors while solving.
The answer shows that the student used problem-solving skills and reasoning.
All work is shown or explained.
1
The answer is partially correct. The student made errors while solving.
The answer shows little problem-solving skills or reasoning.
Some work is shown or explained.
0 The question is not answered.
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To calculate the surface area of a square pyramid, we need to find the area of the base and the area of the four triangles.
The area of the base (which is a square) is calculated by multiplying the base length by the base length:
Base Area = 12 x 12 = 144
To find the area of each triangle, we need to use the formula for the area of a triangle, which is 1/2 * base * height. The base of each triangle is the base length of the pyramid (12), and the height of each triangle is the slant height (18). Since the slant height is the hypotenuse of a right triangle formed by cutting the pyramid in half through the center of the base, we can use the Pythagorean theorem to find the height of each triangle:
height = sqrt(18^2 - 6^2) = sqrt(324 - 36) = sqrt(288) = 12√2
Now, we can calculate the area of each triangle:
Triangle Area = 1/2 * 12 * 12√2 = 72√2
There are 4 triangles, so the total area of the triangles is:
4 * 72√2 = 288√2
Finally, the total surface area of the square pyramid is the sum of the base area and the triangle areas:
Surface Area = Base Area + Triangle Area = 144 + 288√2
Therefore, the surface area of the square pyramid is 144 + 288√2.