To find the surface area of a pyramid with a square base, we can use the formula:
\[ \text{Surface Area} = \text{Base Area} + \text{Lateral Area} \]
Step 1: Calculate the base area
The base is a square, and the area of a square is given by the formula:
\[ \text{Area} = \text{side}^2 \]
Given that the side of the square base is 3 inches:
\[ \text{Base Area} = 3 , \text{in} \times 3 , \text{in} = 9 , \text{in}^2 \]
Step 2: Calculate the lateral area
The lateral area consists of the areas of the triangular faces of the pyramid. A pyramid with a square base has 4 triangular faces. The area of each triangular face can be computed as:
\[ \text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height} \]
In this case, the base of each triangle is the side of the square base (3 in), and the height of the triangle is the slant height of the pyramid (5 in):
\[ \text{Area of one triangle} = \frac{1}{2} \times 3 , \text{in} \times 5 , \text{in} = \frac{15}{2} , \text{in}^2 = 7.5 , \text{in}^2 \]
Since there are 4 triangular faces, the total lateral area is:
\[ \text{Lateral Area} = 4 \times 7.5 , \text{in}^2 = 30 , \text{in}^2 \]
Step 3: Calculate the total surface area
Now, we can combine the base area and the lateral area:
\[ \text{Surface Area} = \text{Base Area} + \text{Lateral Area} = 9 , \text{in}^2 + 30 , \text{in}^2 = 39 , \text{in}^2 \]
Thus, the surface area of the pyramid is:
\[ \boxed{39 , \text{in}^2} \]
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