To solve the problem, we need to find the area of the shaded regions. Here's the breakdown:

Step 1: Determine the total area of the semicircle

The semicircle has a radius of $4 \, \text{cm}$, as given. The formula for the area of a semicircle is: $\text{Area of semicircle} = \frac{1}{2} \pi r^2$ Substituting $r = 4$: $\text{Area of semicircle} = \frac{1}{2} \pi (4)^2 = \frac{1}{2} \pi \cdot 16 = 8\pi$

Step 2: Calculate the area of the triangle $\triangle MLN$

The triangle $\triangle MLN$ is isosceles with its base $LN$ and height $OM$ (which is the radius). The angle at $O$ (central angle) is $90^\circ$, so the triangle splits into two right triangles with angles $45^\circ$.

Each right triangle has:

• Base = $4 \, \text{cm}$ (half of the semicircle radius),
• Height = $4 \, \text{cm}$.

The area of one of these right triangles is: $\text{Area of one right triangle} = \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot 4 \cdot 4 = 8 \, \text{cm}^2$

Since there are two identical right triangles: $\text{Area of } \triangle MLN = 2 \cdot 8 = 16 \, \text{cm}^2$

Step 3: Find the area of the shaded regions

The shaded regions are the parts of the semicircle excluding $\triangle MLN$. Thus: $\text{Area of shaded regions} = \text{Area of semicircle} - \text{Area of } \triangle MLN$ Substituting: $\text{Area of shaded regions} = 8\pi - 16 \, \text{cm}^2$

Final Answer:

The area of the shaded regions is: $\boxed{8\pi - 16 \, \text{cm}^2}$

Do you have any questions or would you like a more detailed explanation of any step?

Related Questions:

1. How do you calculate the area of a full circle instead of a semicircle?
2. Why does the triangle split into two 45°-45°-90° triangles in this problem?
3. What is the significance of expressing answers in terms of $\pi$?
4. How can we verify the dimensions of the triangle using trigonometry?
5. If the radius was doubled, how would the shaded area change?

Tip:

Always label all given dimensions and angles clearly in geometric problems to avoid confusion during calculations!