1. **State the problem:** Calculate the area of a trapezium with top base $12$ cm, bottom base $6$ cm, and height $5$ cm. 2. **Formula for the area of a trapezium:** $$\text{Area} = \frac{(a + b)}{2} \times h$$ where $a$ and $b$ are the lengths of the two parallel sides (bases), and $h$ is the height. 3. **Substitute the given values:** $$a = 12, \quad b = 6, \quad h = 5$$ 4. **Calculate the sum of the bases:** $$a + b = 12 + 6 = 18$$ 5. **Apply the formula:** $$\text{Area} = \frac{18}{2} \times 5$$ 6. **Simplify the fraction:** $$\text{Area} = \cancel{\frac{18}{2}}^{9} \times 5$$ 7. **Multiply to find the area:** $$\text{Area} = 9 \times 5 = 45$$ 8. **Final answer:** The area of the trapezium is $45$ square centimeters.