1. **State the problem:** We need to find the area of a right-angled triangle with base $7$ cm and height $6$ cm. 2. **Formula for the area of a triangle:** $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ 3. **Apply the formula:** $$\text{Area} = \frac{1}{2} \times 7 \times 6$$ 4. **Calculate the multiplication:** $$\text{Area} = \frac{1}{2} \times 42$$ 5. **Simplify the fraction:** $$\text{Area} = \cancel{\frac{1}{2}} \times \cancel{42} = 21$$ 6. **Final answer:** The area of the triangle is $21$ square centimeters.