To calculate the surface area of a square pyramid, we need to consider both the area of the base and the area of the four triangular faces.
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Area of the Base: The base is a square with side length \( s = 20 \) cm. \[ \text{Area of the base} = s^2 = 20 , \text{cm} \times 20 , \text{cm} = 400 , \text{cm}^2 \]
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Area of the Triangular Faces: Each triangular face has a base equal to the side of the square and a height that needs to be calculated.
The height of the triangular face (slant height \( l \)) can be found using the Pythagorean theorem. The height of the pyramid (vertical height) is \( h = 26 \) cm, and half the base edge length (to form a right triangle) is \[ \frac{s}{2} = \frac{20}{2} = 10 , \text{cm}. \]
Using the Pythagorean theorem: \[ l = \sqrt{h^2 + \left(\frac{s}{2}\right)^2} = \sqrt{26^2 + 10^2} = \sqrt{676 + 100} = \sqrt{776} \approx 27.86 , \text{cm}. \]
The area of one triangular face is: \[ \text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 20 , \text{cm} \times l = \frac{1}{2} \times 20 , \text{cm} \times 27.86 , \text{cm} \approx 277.6 , \text{cm}^2. \]
Thus, the area of all four triangular faces is: \[ \text{Total area of triangles} = 4 \times 277.6 , \text{cm}^2 \approx 1110.4 , \text{cm}^2. \]
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Total Surface Area: Finally, the total surface area of the square pyramid is the sum of the base area and the area of the triangular faces: \[ \text{Surface Area} = \text{Area of the base} + \text{Total area of triangles} = 400 , \text{cm}^2 + 1110.4 , \text{cm}^2 \approx 1510.4 , \text{cm}^2. \]
Rounding to the nearest whole number, the surface area is approximately: \[ \text{Surface Area} \approx 1510 , \text{cm}^2. \]
Looking at the available options, the closest answer is \( \boxed{1514 , \text{cm}^2} \).