The problem given asks us to calculate the area of the portion of an equilateral triangle that lies inside a circle, where:

• The triangle $ABC$ is equilateral, and each side is 4 cm.
• A circle is drawn with diameter $BC$.
• We need to find the area of the portion of the triangle that lies inside the circle.

Step 1: Geometry of the Triangle and Circle

Since triangle $ABC$ is equilateral, all its sides are equal to 4 cm. The circle is drawn with diameter $BC$, meaning the diameter of the circle is 4 cm.

Radius of the Circle:

The radius $r$ of the circle is half the diameter: $r = \frac{BC}{2} = \frac{4}{2} = 2 \, \text{cm}$

Step 2: Area of the Triangle $ABC$

The area $A$ of an equilateral triangle with side length $s$ is given by the formula: $A = \frac{\sqrt{3}}{4} \cdot s^2$ Substituting $s = 4$: $A_{\text{triangle}} = \frac{\sqrt{3}}{4} \cdot 4^2 = \frac{\sqrt{3}}{4} \cdot 16 = 4\sqrt{3} \, \text{cm}^2$

Step 3: Circle Area and Relevant Sector

Since the circle is drawn with diameter $BC$, it encloses part of the triangle. The total area of the circle is: $A_{\text{circle}} = \pi r^2 = \pi \cdot 2^2 = 4\pi \, \text{cm}^2$ The relevant portion of the circle is a sector that corresponds to the angle subtended by vertex $A$, which is $60^\circ$ (since the triangle is equilateral).

The area of a sector is given by: $A_{\text{sector}} = \frac{\theta}{360^\circ} \cdot A_{\text{circle}} = \frac{60^\circ}{360^\circ} \cdot 4\pi = \frac{1}{6} \cdot 4\pi = \frac{2\pi}{3} \, \text{cm}^2$

Step 4: Area Inside the Circle

The area inside the circle, which is a portion of the triangle plus the sector of the circle, can be calculated by combining the areas of the triangle and the sector: $A_{\text{portion}} = A_{\text{triangle}} + A_{\text{sector}} = 4\sqrt{3} + \frac{2\pi}{3}$

Final Answer:

Thus, the area of the portion of the triangle that lies inside the circle is: $\boxed{2( \sqrt{3} + \frac{2\pi}{3})}$ This corresponds to option (1).

Would you like any additional details or have any questions?

Here are five related questions for further understanding:

1. How would the answer change if the triangle were not equilateral?
2. What is the formula for the area of a sector of a circle?
3. How can we generalize this result for a triangle with a different side length?
4. How would you calculate the area of the triangle outside the circle?
5. How would you handle this problem if the diameter of the circle was not one of the triangle’s sides?

Tip: Remember that in geometric problems involving circles and polygons, symmetry and known angle measures can simplify area calculations.