The problem involves a large circle containing three semicircles placed across its diameter, and you are asked to find the shaded area. Here's how we can solve it step by step:

Step 1: Identify Key Information

• Radius of each small semicircle: 2 cm.
• Since three semicircles are placed across the diameter, we infer that the diameter of the large circle must be equal to the sum of the diameters of the three semicircles.

Each semicircle has a diameter of 4 cm (since the radius is 2 cm), so the total diameter of the large circle is: $\text{Total diameter of large circle} = 4 \, \text{cm} + 4 \, \text{cm} + 4 \, \text{cm} = 12 \, \text{cm}.$

The radius of the large circle is half of this: $\text{Radius of large circle} = \frac{12}{2} = 6 \, \text{cm}.$

Step 2: Calculate the Area of the Large Circle

The area of a circle is given by: $\text{Area} = \pi r^2.$ Substituting the radius of the large circle: $\text{Area of large circle} = \pi \times 6^2 = 36\pi \, \text{cm}^2.$

Step 3: Calculate the Area of the Three Semicircles

Each semicircle has a radius of 2 cm, so the area of one semicircle is: $\text{Area of one semicircle} = \frac{1}{2} \times \pi r^2 = \frac{1}{2} \times \pi \times 2^2 = 2\pi \, \text{cm}^2.$ For three semicircles: $\text{Total area of three semicircles} = 3 \times 2\pi = 6\pi \, \text{cm}^2.$

Step 4: Find the Shaded Area

The shaded area is the area of the large circle minus the area occupied by the three semicircles: $\text{Shaded area} = 36\pi - 6\pi = 30\pi \, \text{cm}^2.$ Thus, the shaded area is approximately: $\text{Shaded area} \approx 30 \times 3.1416 = 94.25 \, \text{cm}^2.$

Final Answer:

The shaded area is 94.25 cm².

Would you like a breakdown of any part of this solution?

Here are 5 related questions you might want to explore:

1. How would the problem change if the radius of the semicircles were different?
2. Can you calculate the circumference of the large circle?
3. What if there were four semicircles instead of three—how would that affect the total shaded area?
4. How would you find the area if the semicircles were positioned differently, say along a curved line?
5. Can you calculate the area of each unshaded region separately?

Tip:

Whenever dealing with composite shapes, it's helpful to break down the problem into familiar geometric areas (like circles and semicircles) and sum or subtract them accordingly.