How do you calculate the lateral area and sueface area of a pyramid whose height is 8ft and whose base is a square with 12 sides.
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To calculate the lateral area and surface area of a pyramid, you need to know the height, base shape, and side length. In this case, you mentioned that the height of the pyramid is 8ft and the base is a square with 12 sides. However, a square cannot have 12 sides, so we'll assume that you mean a regular square pyramid with a base edge length of 12ft (since a regular square pyramid has a square base).
To calculate the lateral area of a pyramid, you need to find the area of all the triangular faces that make up the sides of the pyramid. In a regular square pyramid, all the triangular faces are congruent.
To find the lateral area, you can follow these steps:
1. Calculate the slant height (s) of the pyramid using the Pythagorean theorem: s = √(h² + (e/2)²), where h is the height and e is the edge length of the base.
=> In this case, h = 8ft and e = 12ft.
=> s = √(8² + (12/2)²) = √(64 + 36) = √100 = 10ft.
2. Calculate the lateral area (A) of one triangular face using the formula: A = (1/2) * base * height, where the base is the edge length of the square and the height is the slant height.
=> In this case, the base of the triangular face is 12ft and the height is 10ft.
=> A = (1/2) * 12ft * 10ft = 60ft².
3. Multiply the lateral area of one triangular face by 4 (since all the triangular faces are congruent in a regular square pyramid) to find the total lateral area.
=> Total Lateral Area = 4 * 60ft² = 240ft².
Now, let's calculate the surface area (including the base) of the pyramid.
4. Calculate the base area of the pyramid. Since the base is a square, the area can be found by multiplying the length of one side by itself.
=> Base Area = 12ft * 12ft = 144ft².
5. Add the lateral area to the base area to get the total surface area of the pyramid.
=> Total Surface Area = Lateral Area + Base Area = 240ft² + 144ft² = 384ft².
Therefore, the lateral area of the pyramid is 240ft² and the total surface area is 384ft².