Let's solve each part step-by-step, focusing on finding the area of each shaded region.

Part (a)

1. The large circle has two smaller circles of radius $2 \, \text{cm}$ inside it.
2. The radius of the large circle is also $2 \, \text{cm} + 2 \, \text{cm} = 4 \, \text{cm}$.
3. Calculate the area of the large circle: $\text{Area of large circle} = \pi \times (4)^2 = 16\pi$
4. Calculate the area of each small circle: $\text{Area of each small circle} = \pi \times (2)^2 = 4\pi$
5. Subtract the area of the two smaller circles from the area of the large circle to get the shaded area: $\text{Shaded area} = 16\pi - 2 \times 4\pi = 16\pi - 8\pi = 8\pi$

Part (b)

1. Here, we have a semicircle of radius $4 \, \text{cm}$.
2. Calculate the area of the semicircle: $\text{Area of semicircle} = \frac{1}{2} \pi \times (4)^2 = \frac{1}{2} \pi \times 16 = 8\pi$
3. Inside the semicircle, there is a smaller circle with radius $2 \, \text{cm}$.
4. Calculate the area of the small circle: $\text{Area of small circle} = \pi \times (2)^2 = 4\pi$
5. Subtract the area of the small circle from the area of the semicircle to get the shaded area: $\text{Shaded area} = 8\pi - 4\pi = 4\pi$

Part (c)

1. The large circle has a radius of $5 \, \text{cm}$.
2. Calculate the area of the large circle: $\text{Area of large circle} = \pi \times (5)^2 = 25\pi$
3. Inside the large circle, there is a smaller semicircle with radius $5 \, \text{cm} / 2 = 2.5 \, \text{cm}$.
4. Calculate the area of the small semicircle: $\text{Area of small semicircle} = \frac{1}{2} \pi \times (2.5)^2 = \frac{1}{2} \pi \times 6.25 = 3.125\pi$
5. Subtract the area of the small semicircle from the area of the large circle to get the shaded area: $\text{Shaded area} = 25\pi - 3.125\pi = 21.875\pi$

Part (d)

1. The square has a side length of $30 \, \text{cm}$.
2. Calculate the area of the square: $\text{Area of square} = 30 \times 30 = 900$
3. The square is divided into four quarter circles, each with a radius of $15 \, \text{cm}$ (half the side length).
4. Calculate the area of each quarter circle: $\text{Area of each quarter circle} = \frac{1}{4} \pi \times (15)^2 = \frac{1}{4} \pi \times 225 = 56.25\pi$
5. The total area of the four quarter circles is: $4 \times 56.25\pi = 225\pi$
6. Subtract the area of the four quarter circles from the area of the square to get the shaded area: $\text{Shaded area} = 900 - 225\pi$

Let me know if you'd like any additional details on these steps or have further questions.

Follow-up Questions

1. How is the area of a semicircle derived from a full circle's area?
2. What happens to the shaded area if the radius of the smaller circles in (a) changes?
3. Can you calculate the circumference of each shape given the radius or diameter?
4. How would the shaded area change in (d) if it only contained two quarter circles?
5. How do you calculate the area of complex shapes made up of multiple geometric shapes?

Quick Tip

When working with composite areas, it's often helpful to break down complex shapes into basic geometric figures, calculate each area individually, and then sum or subtract as needed.