Calculate the surface area of the pyramid.

The base has a side length of $5 \text{ cm}$.

$$\text{Area of base} = \text{side} \times \text{side} = (5 \text{ cm})^2 = 25 \text{ cm}^2$$

Each triangular face has a base equal to the side length of the square base ($5 \text{ cm}$) and a perpendicular height of $4 \text{ cm}$.

$$\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 5 \text{ cm} \times 4 \text{ cm} = 10 \text{ cm}^2$$

Since there are four identical triangular faces:

$$\text{Total area of triangles} = 4 \times \text{Area of one triangle} = 4 \times 10 \text{ cm}^2 = 40 \text{ cm}^2$$

The total surface area is the sum of the area of the base and the total area of the triangular faces.

$$\text{Total surface area} = \text{Area of base} + \text{Total area of triangles} = 25 \text{ cm}^2 + 40 \text{ cm}^2 = 65 \text{ cm}^2$$

The surface area of the pyramid is $\text{65 cm}^2$.