1. **State the problem:** Find the surface area of a square pyramid with base side length $2$ cm and slant height $5$ cm. 2. **Formula:** The surface area $A$ of a square pyramid is given by: $$A = B + L$$ where $B$ is the area of the base and $L$ is the lateral surface area. 3. **Calculate the base area $B$:** Since the base is a square with side length $s=2$ cm, $$B = s^2 = 2^2 = 4 \text{ cm}^2$$ 4. **Calculate the lateral surface area $L$:** The lateral surface area of a square pyramid is: $$L = \frac{1}{2} \times P \times l$$ where $P$ is the perimeter of the base and $l$ is the slant height. 5. **Calculate the perimeter $P$:** $$P = 4 \times s = 4 \times 2 = 8 \text{ cm}$$ 6. **Calculate $L$ using $l=5$ cm:** $$L = \frac{1}{2} \times 8 \times 5 = 4 \times 5 = 20 \text{ cm}^2$$ 7. **Calculate total surface area $A$:** $$A = B + L = 4 + 20 = 24 \text{ cm}^2$$ **Final answer:** The surface area of the square pyramid is $24$ cm².