Calculate the area of the right-angled triangle.

The formula for the area of a triangle is $\frac{1}{2} \times \text{base} \times \text{height}$.
From the diagram, the base is 8 and the height is 7. The $30^\circ$ angle is not consistent with a right triangle having legs 7 and 8, so we will use the given leg lengths for the area calculation.

$$\text{Area} = \frac{1}{2} \times 8 \times 7$$ $$\text{Area} = 4 \times 7$$ $$\text{Area} = 28 \text{ square units}$$

The expression given is $(\sqrt{200})^2 + 20$.
When a square root is squared, the result is the number inside the square root.

$$(\sqrt{200})^2 = 200$$

Now, add 20 to this result:

$$200 + 20 = 220$$

The area of the triangle is $\boxed{\text{28 square units}}$.
The value of the expression is $\boxed{220}$.