Find the lateral area of the square pyramid. I don’t understand this st all the measure ments are 8m 8m 22m pelase walk me through this!
I assume you mean that each side of the base is 8m, and the height is 22m.
If you want useful answers, ask sensible questions.
So, each lateral face is a triangle with base 8
The height of each triangle is the slant height of the pyramid.
If you look from the side, the altitude from the top to the base lands in the middle, 4m from each side. So, the slant height s can be found by
4^2 + s^2 = 22^2
s = 6√13
So, the lateral area is 4 times the area of each triangle:
area = 4(8 * 6√13)/2 = 96√13 m^2
Find the lateral area of the square pyramid.
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To find the lateral area of a square pyramid, you need to calculate the sum of the areas of all the triangular faces that form the sides of the pyramid.
In this case, we know that the base of the pyramid is a square with side lengths of 8m. To find the area of this square base, you can use the formula for the area of a square, which is side length squared. So, the area of the base would be 8m * 8m = 64m².
Next, we need to calculate the area of each triangular face. Since the pyramid is regular, all the triangular faces will have the same dimensions. Let's call the height of the pyramid h and the slant height of each triangular face l.
The slant height can be found by using the Pythagorean theorem, as the slant height, height, and half the length of the base form a right triangle. The length of half the base is 8m/2 = 4m.
So, using the Pythagorean theorem: l² = (4m)² + h².
We also know that the slant height l is 22m, so we can substitute the values into the equation: 22m² = 4m² + h².
Solving for h, we subtract 4m² from both sides and take the square root: h = √(22m² - 4m²) ≈ √(484m² - 16m²) ≈ √468m² ≈ 21.63m (rounded to two decimal places).
Now that we have both the base area and the height of each triangular face, we can calculate the area of each triangular face using the formula for the area of a triangle, which is base times height divided by 2.
For our triangle, the base is the side length of the square, which is 8m, and the height is the height of the pyramid, which is 21.63m.
Therefore, the area of each triangular face is (8m * 21.63m) / 2 = 86.52m².
Since a square pyramid has four triangular faces, the total lateral area is the sum of the areas of all four triangular faces.
Therefore, the lateral area of the square pyramid is 4 * 86.52m² = 346.08m² (rounded to two decimal places).