To find the surface area of a square pyramid, we need to calculate the area of the base and the area of the four triangular sides.
Step 1: Find the area of the base. The base is a square, and the area of a square is calculated using the formula:
\[ \text{Area of the base} = \text{side} \times \text{side} \]
The side length of the base is 12:
\[ \text{Area of the base} = 12 \times 12 = 144 \]
Step 2: Find the area of one triangular side. The area of a triangle is given by the formula:
\[ \text{Area of a triangle} = \frac{1}{2} \times \text{base} \times \text{height} \]
For the triangles, the base is the same as the side of the square, which is 12, and the height is the slant height, which is 18:
\[ \text{Area of one triangle} = \frac{1}{2} \times 12 \times 18 \]
Calculating this gives:
\[ \text{Area of one triangle} = \frac{1}{2} \times 12 \times 18 = 6 \times 18 = 108 \]
Step 3: Find the total area of the four triangular sides. Since there are four triangular sides, we multiply the area of one triangle by 4:
\[ \text{Total area of triangular sides} = 4 \times 108 = 432 \]
Step 4: Calculate the total surface area. Now, we add the area of the base to the total area of the triangular sides:
\[ \text{Total surface area} = \text{Area of the base} + \text{Total area of triangular sides} \] \[ \text{Total surface area} = 144 + 432 = 576 \]
So, the surface area of the square pyramid is 576 square units.