1. **State the problem:** Calculate the area of a right triangle with legs measuring 5 cm and 8 cm. 2. **Formula:** The area $A$ of a right triangle is given by $$A = \frac{1}{2} \times \text{base} \times \text{height}$$ where the base and height are the two legs that form the right angle. 3. **Apply the formula:** Here, base = 8 cm and height = 5 cm. 4. **Calculate:** $$A = \frac{1}{2} \times 8 \times 5 = \frac{1}{2} \times 40 = 20$$ 5. **Conclusion:** The area of the triangle is $20$ square centimeters.