To find the surface area of a rectangular pyramid, we need to know the area of the base and the lateral surface area (the area of the four triangular sides).
-
Base Area: For a rectangular pyramid with a base that is a rectangle, the area of the base (A_base) can be calculated as: \[ A_{base} = \text{length} \times \text{width} \] Assuming the base is a square (as no specific length and width are given), the base area would be: \[ A_{base} = 6 , \text{cm} \times 6 , \text{cm} = 36 , \text{cm}^2 \]
If we take it as a rectangular base of length 6 cm and width x cm (you need to specify x), the base area would just be \(6 \times x\).
-
Lateral Surface Area: The lateral surface area consists of 4 triangular faces.
Each triangular face has a base equal to the side of the base rectangle and a height that can be calculated. The height of the triangle is the slant height (l), which can be found using the Pythagorean theorem: \[ l = \sqrt{\left(\frac{\text{width}}{2}\right)^2 + h^2} \] Here, \(h\) is the height of the pyramid, which is given as 8 cm.
If we assume a square base again, the width is also 6 cm, therefore: \[ l = \sqrt{\left(\frac{6}{2}\right)^2 + 8^2} = \sqrt{3^2 + 8^2} = \sqrt{9 + 64} = \sqrt{73} \]
For each triangular face, the area (A_triangle) can be calculated as: \[ A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times l \] Since there are 4 triangles: \[ A_{lateral} = 4 \times \frac{1}{2} \times 6 \times \sqrt{73} = 12 \sqrt{73} , \text{cm}^2 \]
-
Total Surface Area: The total surface area (A_total) is given by the sum of the base area and the lateral area: \[ A_{total} = A_{base} + A_{lateral} = 36 + 12\sqrt{73} , \text{cm}^2 \]
To give a numerical answer, we approximate \(\sqrt{73} \approx 8.54\): \[ A_{lateral} \approx 12 \times 8.54 \approx 102.48 , \text{cm}^2 \] Therefore, \[ A_{total} \approx 36 + 102.48 = 138.48 , \text{cm}^2 \]
Thus, the total surface area of the pyramid is approximately \(138.48 , \text{cm}^2\).
Note: If the base is not square, please specify the dimensions or ratio of the rectangular base for a more accurate configuration.