To calculate the area of trapezium $H$, let's analyze the given diagram step by step.

1. The trapezium consists of a larger triangle (colored pink, representing the area of $H$) and a smaller triangle (colored blue) with a given area of $30 \, \text{cm}^2$.

2. The total height of both triangles combined is $27 + 9 = 36 \, \text{cm}$. The smaller triangle has a height of $9 \, \text{cm}$.

3. The formula for the area of a triangle is: $\text{Area of a triangle} = \frac{1}{2} \times \text{base} \times \text{height}$

4. From the smaller triangle, we can determine its base since the area and height are given: $30 = \frac{1}{2} \times \text{base} \times 9$ Solving for the base: $\text{base} = \frac{30 \times 2}{9} = \frac{60}{9} = 6.67 \, \text{cm}$

5. Since the trapezium shares the same base as the larger triangle, the base of trapezium $H$ is $6.67 \, \text{cm}$, and its height is $27 \, \text{cm}$.

6. Using the area formula for a triangle again to find the area of trapezium $H$: $\text{Area of trapezium } H = \frac{1}{2} \times 6.67 \times 27 = 90 \, \text{cm}^2$

Therefore, the area of trapezium $H$ is $90 \, \text{cm}^2$.

Would you like a detailed breakdown of any step? Let me know if you have any further questions!

Here are 5 related questions:

1. How do you calculate the area of a trapezium if both bases are known?
2. What is the relationship between the area of triangles and trapeziums?
3. How can you calculate the height of a triangle given its area and base?
4. How would the problem change if the base of the trapezium were doubled?
5. Can you calculate the perimeter of trapezium $H$ given the current information?

Tip: When working with geometric shapes, always check if they share common sides or dimensions to simplify your calculations.