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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)
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Bot GPT 3.5
answered
To calculate the surface area of a square pyramid, we need to find the area of the base and the lateral faces.
1. Area of the base:
Since the base of the square pyramid is a square, we can use the formula for finding the area of a square:
Base area = length * width
Base area = 12 * 12
Base area = 144 square units
2. Area of the four triangular faces (lateral faces):
To find the area of each triangular face, we use the formula for the area of a triangle:
Area = 1/2 * base * height
Given that the slant height is 18 and the base length is 12, we can use the Pythagorean theorem to find the height of each triangular face.
We can see that the height forms a right triangle with the base and the slant height, so we can use the Pythagorean theorem:
height^2 + (1/2 * 12)^2 = 18^2
height^2 + 36 = 324
height^2 = 288
height = √288
height ≈ 16.97
Area of each triangular face = 1/2 * 12 * 16.97
Area of each triangular face ≈ 101.82 square units
Since there are 4 triangular faces: Total lateral surface area = 4 * 101.82 = 407.28 square units
3. Total surface area of the square pyramid:
Surface area = Base area + Lateral surface area
Surface area = 144 + 407.28
Surface area = 551.28 square units
Therefore, the surface area of the square pyramid is 551.28 square units.
dale
answered
can you make it smaller
Bot GPT 3.5
answered
Sure! Here is a more concise version of the calculation:
1. Base Area = 12 * 12 = 144 square units
2. Height of each triangular face = √(18^2 - 6^2) = √(324 - 36) = √288 ≈ 16.97
3. Area of each triangular face = 1/2 * 12 * 16.97 ≈ 101.82 square units
4. Lateral surface area = 4 * 101.82 = 407.28 square units
5. Total surface area = Base area + Lateral surface area = 144 + 407.28 = 551.28 square units
Therefore, the surface area of the square pyramid is approximately 551.28 square units.
dale
answered
can you make it more simple
Bot GPT 3.5
answered
Absolutely! Here's a simpler breakdown of the calculation:
1. Base Area = 144 square units (12 * 12)
2. Height of each triangular face = 16.97 (approximate)
3. Area of each triangular face ≈ 102 square units (1/2 * 12 * 16.97)
4. Lateral surface area = 4 * 102 = 408 square units (approximate)
5. Total surface area = Base area + Lateral surface area = 144 + 408 = 552 square units
Therefore, the surface area of the square pyramid is approximately 552 square units.
dale
answered
what does * mean
thabk you
Bot GPT 3.5
answered
In mathematics, the symbol "*" represents multiplication. When you see an expression like 12 * 12, it means you are multiplying 12 by 12 to get a result. The "*" symbol is commonly used to indicate multiplication in arithmetic and algebraic expressions.
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