Find the surface area of a right rectangular pyramid with a base 10 in by 8 in and a height of 10 in. (Answer to the nearest square inch)
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To find the surface area of a right rectangular pyramid, we need to calculate the area of the base and the area of the lateral faces, and then add them together.
1. Calculate the area of the base:
The base of the pyramid is a rectangle with dimensions of 10 inches by 8 inches. The formula to calculate the area of a rectangle is length × width. Therefore, the area of the base is 10 in × 8 in = 80 square inches.
2. Calculate the area of the lateral faces:
The lateral faces of a right rectangular pyramid are triangles. To find the area of a triangle, we need the base and the height of the triangle. In this case, the base of each triangular face is the same as one of the sides of the rectangle, which is 8 inches. The height of each triangular face is the slant height of the pyramid, which can be found using the Pythagorean theorem.
To find the slant height, we can draw a right triangle with the height, the base, and the slant height as its sides. The height of the pyramid is given as 10 inches, and we can use the Pythagorean theorem to find the slant height.
By considering the triangle formed by the height, the slant height, and the base, we can write the equation: h^2 + (base/2)^2 = slant height^2
Plugging in the given values, we get: 10^2 + (8/2)^2 = slant height^2
Simplifying: 100 + 16 = slant height^2
Calculating further: 116 = slant height^2
Taking the square root of both sides: slant height ≈ 10.77 inches (rounded to two decimal places)
Now that we have the slant height, we can calculate the area of each triangular lateral face. The formula for the area of a triangle is base × height / 2.
Each triangular lateral face will have a base of 8 inches and a height of 10.77 inches. Therefore, the area of each triangular lateral face is (8 in × 10.77 in) / 2 = 43.08 square inches (rounded to two decimal places).
3. Calculate the total surface area:
The total surface area of the rectangular pyramid is the sum of the area of the base and the area of the four triangular lateral faces.
Total surface area = area of base + area of lateral faces
Total surface area = 80 square inches + (4 × 43.08 square inches) = 80 square inches + 172.32 square inches = 252.32 square inches.
Therefore, the surface area of the right rectangular pyramid is approximately 252.32 square inches (rounded to the nearest square inch).