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A pyramid on its side is shown with its height measuring 22 m and its base measuring 8 m.
Find the lateral area of the square pyramid.
The lateral area of a pyramid is given by the formula:
Lateral area = (1/2) * perimeter of base * slant height
To find the perimeter of the base, we need to know the length of one side of the square base. Since the base measures 8 m, each side of the square base measures:
8 / √2 ≈ 5.66 m
To find the slant height, we can use the Pythagorean theorem. The slant height is the hypotenuse of a right triangle with one leg equal to half the length of a side of the base (since the pyramid is on its side) and the other leg equal to the height of the pyramid. Thus:
slant height = √[(1/2 * 8)^2 + 22^2] ≈ 22.63 m
Now we can plug in these values to find the lateral area:
Lateral area = (1/2) * 4 * 5.66 * 22.63 ≈ 254.5 m^2
Therefore, the lateral area of the square pyramid is approximately 254.5 square meters.
To find the lateral area of a square pyramid, we need to calculate the sum of the areas of all the triangular faces that make up its sides.
Step 1: Find the slant height of the pyramid
The slant height can be found using the Pythagorean theorem. The slant height (l) is the hypotenuse of a right triangle with one leg as the height of the pyramid (h) and the other leg as half the length of one side of the base (s/2). We can substitute the given values to find l.
Using the Pythagorean theorem: l^2 = (s/2)^2 + h^2
Plugging in the values: l^2 = (8/2)^2 + 22^2
Simplifying: l^2 = 4^2 + 22^2
Calculating: l^2 = 16 + 484
Calculating: l^2 = 500
Taking the square root: l ≈ √500
Calculating: l ≈ 22.36
Step 2: Calculate the area of each triangular face
The area of a triangle can be calculated using the formula: Area = 1/2 * base * height.
In this case, the base of each triangular face is the length of one side of the base (8m), and the height is the slant height (22.36m).
Calculating: Area = 1/2 * 8 * 22.36
Calculating: Area ≈ 88.72m^2
Step 3: Calculate the total lateral area
Since there are four triangular faces in a square pyramid, we need to multiply the area of each triangular face by the number of faces.
Calculating: Total Lateral Area = 88.72m^2 * 4
Calculating: Total Lateral Area ≈ 354.88m^2
Therefore, the lateral area of the square pyramid is approximately 354.88 square meters.
To find the lateral area of a square pyramid, you need to find the perimeter of the base and multiply it by half of the slant height.
Step 1: Find the perimeter of the base
Since the base of the pyramid is a square, all sides are equal. The formula to find the perimeter of a square is P = 4s, where P is the perimeter and s is the length of a side.
In this case, the length of each side of the base is 8 m. Therefore, the perimeter of the base is:
P = 4 * 8 = 32 m
Step 2: Find the slant height
The slant height of a pyramid is a line from the apex (top) to the edge of the base triangle. To find the slant height, we will use the Pythagorean theorem, which states that the square of the hypotenuse (slant height) is equal to the sum of the squares of the other two sides. In this case, the two sides are half of the height (11 m) and half of the base (4 m).
Using the Pythagorean theorem, we can calculate the slant height as follows:
slant height^2 = (half of the height)^2 + (half of the base)^2
slant height^2 = 11^2 + 4^2
slant height^2 = 121 + 16
slant height^2 = 137
slant height ≈ √137 ≈ 11.70 m
Step 3: Calculate the lateral area
The lateral area of a square pyramid is given by the formula:
Lateral area = (perimeter of the base) * (slant height) / 2
Lateral area = 32 m * 11.70 m / 2
Lateral area ≈ 375.04 m^2
Therefore, the lateral area of the square pyramid is approximately 375.04 m^2.