Area of Circular Segment Calculation

To find the area of the segment formed by a chord in a circle, we calculate the area of the sector defined by the central angle and subtract the area of the triangle formed by the chord and the radii to the endpoints of the chord.

Formulas Used

• Area of Sector = $\frac{\theta}{360^{\circ}} \times \pi r^2$
• Area of Triangle = $\frac{1}{2} r^2 \sin(\theta)$
• Area of Segment = Area of Sector - Area of Triangle

Given Values

• Radius ($r$): 6 cm
• Central Angle ($\theta$): $120^{\circ}$

Step 1: Calculate the Area of the Sector

Using the formula for the area of a sector:

Area of Sector = $\frac{120^{\circ}}{360^{\circ}} \times \pi (6 \text{ cm})^2$

Area of Sector = $\frac{1}{3} \times \pi \times 36 \text{ cm}^2$

Area of Sector = $12\pi \text{ cm}^2$

Step 2: Calculate the Area of the Triangle

Using the formula for the area of a triangle with two sides and the included angle:

Area of Triangle = $\frac{1}{2} (6 \text{ cm})^2 \sin(120^{\circ})$

We know that $\sin(120^{\circ}) = \frac{\sqrt{3}}{2}$.

Area of Triangle = $\frac{1}{2} \times 36 \text{ cm}^2 \times \frac{\sqrt{3}}{2}$

Area of Triangle = $18 \times \frac{\sqrt{3}}{2} \text{ cm}^2$

Area of Triangle = $9\sqrt{3} \text{ cm}^2$

Step 3: Calculate the Area of the Segment

Subtract the area of the triangle from the area of the sector:

Area of Segment = Area of Sector - Area of Triangle

Area of Segment = $(12\pi - 9\sqrt{3}) \text{ cm}^2$

The area of the segment is $12\pi - 9\sqrt{3} \text{ cm}^2$. This corresponds to Option A.