Calculate the surface area of the square pyramid that has a base length of 12 and a slant height of 18. Show your work.?????????????
To calculate the surface area of a square pyramid, we need to add the area of the base to the sum of the areas of the four triangular faces.
The area of the base of the square pyramid is found by multiplying the base length by itself:
Base area = 12 * 12 = 144
Next, we need to calculate the area of the four triangular faces. Each triangular face is an isosceles triangle with a base length of 12 and a slant height of 18. We can calculate the height of each triangle using the Pythagorean theorem:
height = sqrt(18^2 - (12/2)^2) = sqrt(324 - 36) = sqrt(288) = 12√2
Now, we can calculate the area of one triangle:
Triangle area = (1/2) * base length * height = (1/2) * 12 * 12√2 = 72√2
Since there are four triangular faces on a square pyramid, we need to calculate the total area of all the faces:
Total face area = 4 * 72√2 = 288√2
Finally, we can find the total surface area of the square pyramid by adding the base area and the face area:
Surface area = base area + total face area = 144 + 288√2 ≈ 588.90
Therefore, the surface area of the square pyramid with a base length of 12 and a slant height of 18 is approximately 588.90.
neater
Sure, here is a neater presentation of the solution:
Given:
Base length = 12
Slant height = 18
1. Calculate base area:
Base area = 12 * 12 = 144
2. Calculate the height of each triangular face:
Height = sqrt(18^2 - (12/2)^2) = sqrt(324 - 36) = sqrt(288) = 12√2
3. Calculate the area of one triangular face:
Triangle area = (1/2) * 12 * 12√2 = 72√2
4. Calculate the total face area:
Total face area = 4 * 72√2 = 288√2
5. Calculate the total surface area:
Surface area = base area + total face area
Surface area = 144 + 288√2 ≈ 588.90
Therefore, the surface area of the square pyramid with a base length of 12 and a slant height of 18 is approximately 588.90.