To solve for the area of the sector $OABC$, let's proceed step-by-step.

Step 1: Analyze the given information

• $OA = OB = 12 \, \text{cm}$: These are the radii of the circle forming the sector $OABC$.
• $AB = 12\sqrt{3} \, \text{cm}$: The chord of the sector.

We need to calculate the area of the sector $OABC$.

Step 2: Find the central angle of the sector

The central angle $\theta$ can be determined using the cosine rule in triangle $OAB$: $\cos\theta = \frac{OA^2 + OB^2 - AB^2}{2 \cdot OA \cdot OB}$

Substitute the given values: $OA = OB = 12, \quad AB = 12\sqrt{3}$ $\cos\theta = \frac{12^2 + 12^2 - (12\sqrt{3})^2}{2 \cdot 12 \cdot 12}$ Simplify: $\cos\theta = \frac{144 + 144 - 432}{288}$ $\cos\theta = \frac{-144}{288} = -\frac{1}{2}$

From trigonometric principles, $\cos\theta = -\frac{1}{2}$ implies: \theta = 120^\circ \quad \text{(in radians, \theta = \frac{2\pi}{3})}.

Step 3: Area of the sector

The formula for the area of a sector is: $\text{Area of sector} = \frac{\theta}{2\pi} \cdot \pi r^2$

Substitute $\theta = \frac{2\pi}{3}$ and $r = 12$: $\text{Area of sector} = \frac{\frac{2\pi}{3}}{2\pi} \cdot \pi (12)^2$ Simplify: $\text{Area of sector} = \frac{2}{3} \cdot 144\pi = 96\pi \, \text{cm}^2.$

Final Answer:

The area of the sector $OABC$ is: $\boxed{96\pi \, \text{cm}^2.}$

Let me know if you'd like a detailed explanation of any step, or have more questions! Here are some related queries:

1. How do we convert degrees to radians in calculations?
2. What is the significance of the cosine rule in geometry?
3. How is the formula for sector area derived from a circle's area?
4. Can the chord length be used to directly find the area of the sector?
5. What are other real-world applications of sectors?

Tip: Always check if the angle is in degrees or radians when calculating circular measures!