To solve the problem of determining the area of the shaded region, let's break it into steps:

Problem Analysis:

1. Given:

   • $DE = DF = 4 \, \text{cm}$ (isosceles triangle).
   • $\angle EDF = 30^\circ$ (central angle of the sector).
   • The shaded region is the area of the triangle minus the area of the circular segment.

2. Steps:

   • Calculate the area of the isosceles triangle $\triangle DEF$.
   • Calculate the area of the circular segment (sector minus triangle within the segment).
   • Subtract the circular segment area from the triangle area.

Solution:

1. Area of the Isosceles Triangle:

• The base angles are equal since $DE = DF$, and each angle is: $\angle EFD = \angle EDF = \frac{180^\circ - 30^\circ}{2} = 75^\circ$
• Height of the triangle from point $D$ to base $EF$: Using trigonometry, the height is: $h = 4 \cdot \sin(75^\circ)$ Approximation: $\sin(75^\circ) = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ \approx 0.9659$, so: $h \approx 4 \cdot 0.9659 = 3.8636 \, \text{cm}$
• Base of triangle $EF$: $EF = 2 \cdot (4 \cdot \cos(75^\circ)) \quad (\cos(75^\circ) \approx 0.2588)$ $EF \approx 2 \cdot 4 \cdot 0.2588 = 2.0704 \, \text{cm}$
• Area of the triangle: $A_{\text{triangle}} = \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot 2.0704 \cdot 3.8636 \approx 3.998 \, \text{cm}^2$

2. Area of the Circular Sector:

• The radius of the circle is $4 \, \text{cm}$.
• Area of the sector: $A_{\text{sector}} = \frac{\theta}{360^\circ} \cdot \pi r^2 = \frac{30^\circ}{360^\circ} \cdot \pi \cdot 4^2$ $A_{\text{sector}} = \frac{1}{12} \cdot 16\pi \approx 4.1888 \, \text{cm}^2$

3. Area of the Circular Segment:

• The area of the circular segment is the sector area minus the triangle area within the sector.
• The triangle within the sector is: $A_{\text{small triangle}} = \frac{1}{2} r^2 \sin(\theta)$ where $\sin(30^\circ) = 0.5$: $A_{\text{small triangle}} = \frac{1}{2} \cdot 4^2 \cdot 0.5 = 4 \, \text{cm}^2$
• Area of the circular segment: $A_{\text{segment}} = A_{\text{sector}} - A_{\text{small triangle}} = 4.1888 - 4 \approx 0.1888 \, \text{cm}^2$

4. Shaded Region:

The shaded area is: $A_{\text{shaded}} = A_{\text{triangle}} - A_{\text{segment}}$ $A_{\text{shaded}} \approx 3.998 - 0.1888 = 3.8092 \, \text{cm}^2$

Final Answer:

The area of the shaded region is approximately: $\boxed{3.81 \, \text{cm}^2}$

Would you like detailed verification of any step or further explanations?

Relative Questions:

1. How does the base-angle relationship in an isosceles triangle affect its height?
2. What is the significance of calculating circular segments for areas involving arcs?
3. Can you derive the formula for the area of a circular segment?
4. How does changing the angle $\theta$ influence the shaded area?
5. What happens to the shaded area if the radius is doubled?

Tip:

When solving geometry problems involving circles and triangles, always separate overlapping regions into distinct, calculable parts like sectors, triangles, or rectangles.