Find the surface area of a rectangular pyramid with these measurements: l = 13 cm, w = 11 cm, and h = 8 cm. Express your answer as a decimal rounded to the nearest hundredth.(1 point)
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To find the surface area of a rectangular pyramid, you need to consider four triangular faces and one rectangular base. The formula for the surface area of a rectangular pyramid is:
Surface Area = Base Area + (1/2) × Perimeter of Base × Slant Height
First, let's find the base area. Since the base of the pyramid is rectangular, the formula for finding the area of a rectangle is:
Area of Rectangle = length × width
Therefore, the base area (BA) is:
BA = 13 cm × 11 cm
Next, we need to find the perimeter of the base. Since the base is rectangular, the formula for finding the perimeter of a rectangle is:
Perimeter of Rectangle = 2 × (length + width)
Therefore, the perimeter of the base (P) is:
P = 2 × (13 cm + 11 cm)
Lastly, we need to find the slant height (L) of the pyramid. To do this, we can use the Pythagorean theorem. The slant height is the hypotenuse of a right-angled triangle formed by one of the triangular faces, the height, and half the base. The formula is:
Slant Height (L) = √(height² + (1/2 × base)²)
Therefore, the slant height (L) is:
L = √(8 cm² + (1/2 × 13 cm)²)
Now, we can plug in the values into the surface area formula:
Surface Area = BA + (1/2) × P × L
Surface Area = (13 cm × 11 cm) + (1/2) × (2 × (13 cm + 11 cm)) × √(8 cm² + (1/2 × 13 cm)²)
Simplifying this equation will give you the surface area of the rectangular pyramid. Round the final result to the nearest hundredth.
Note: Make sure to perform the operations in the correct order (e.g., calculate the slant height first before calculating the surface area).
To find the surface area of a rectangular pyramid, we need to find the area of each face and add them together.
The base of the pyramid is a rectangle with length 13 cm and width 11 cm. The area of the base is given by the formula length × width, so it's 13 cm × 11 cm = 143 cm².
The three triangular faces have the same area. To find the area of one of these faces, we can use the formula 1/2 × base × height, since each face is a right triangle. The base is 11 cm (the width of the base of the pyramid) and the height is 8 cm, so the area of one face is 1/2 × 11 cm× 8 cm = 44 cm².
Since there are three triangular faces, the total area of the three triangular faces is 3 × 44 cm² = 132 cm².
To find the total surface area of the pyramid, we add the area of the base and the area of the three triangular faces: 143 cm² + 132 cm² = 275 cm².
Therefore, the surface area of the rectangular pyramid is 275 cm².
Apologies for the error in my previous response. Let's calculate the correct surface area of the rectangular pyramid.
The base of the pyramid is a rectangle with length 13 cm and width 11 cm. The area of the base is given by the formula length × width, so it's 13 cm × 11 cm = 143 cm².
The three triangular faces have the same area. To find the area of one of these faces, we can use the formula 1/2 × base × height since each face is a right triangle. The base of each triangular face is the same as the width of the base, which is 11 cm. The height of each triangular face, which is also the slant height of the pyramid, can be found using the Pythagorean theorem. The Pythagorean theorem 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 slant height is the hypotenuse, the width of the base is one side, and the height of the pyramid is the other side. So we have:
slant height² = width² + height²
slant height² = 11 cm² + 8 cm²
slant height² = 121 cm² + 64 cm²
slant height² = 185 cm²
To find the slant height, we take the square root of both sides:
slant height = √185 cm ≈ 13.60 cm (rounded to the nearest hundredth)
Now we can calculate the area of one triangular face using 1/2 × base × height:
Area of one triangular face = 1/2 × 11 cm × 13.60 cm ≈ 74.80 cm² (rounded to the nearest hundredth)
Since there are three triangular faces, the total area of the three triangular faces is 3 × 74.80 cm² = 224.40 cm² (rounded to the nearest hundredth).
Therefore, the surface area of the rectangular pyramid is the sum of the area of the base and the area of the three triangular faces:
Surface area = 143 cm² + 224.40 cm² ≈ 367.40 cm² (rounded to the nearest hundredth).
Thus, the surface area of the rectangular pyramid is approximately 367.40 cm².