Calculate the surface area of the square pyramid that has a base length of 12 and slant height of 18

Answer 1

To calculate the surface area of a square pyramid, you need to find the area of the base and the area of each of the four triangular faces.

1. Calculate the area of the base:
Since the base of the pyramid is a square, the area of the base can be found using the formula for the area of a square:
Area = side length^2 = 12^2 = 144 square units

2. Calculate the area of each triangular face:
The slant height (l) of the pyramid is given as 18. The slant height is the hypotenuse of a right triangle formed by the slant height, the height (h) (which is the altitude from the apex of the pyramid to the base), and half of one side of the base (a/2).
Using the Pythagorean theorem, we can find the height (h):
h = sqrt(l^2 - (a/2)^2)
h = sqrt(18^2 - (12/2)^2) = sqrt(324 - 36) = sqrt(288) = 12sqrt(2)

Now, we can calculate the area of each triangular face using the formula for the area of a triangle:
Area = 0.5 * base * height
Area = 0.5 * 12 * 12sqrt(2) = 72sqrt(2) square units

3. Calculate the total surface area of the pyramid:
The total surface area is the sum of the area of the base and the four triangular faces:
Surface Area = Area of Base + 4(Area of Triangular Face)
Surface Area = 144 + 4(72sqrt(2))
Surface Area = 144 + 288sqrt(2)
Surface Area ≈ 573.97 square units

Therefore, the surface area of the square pyramid with a base length of 12 and a slant height of 18 is approximately 573.97 square units.