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To solve the problem, we need to calculate the area of the segment \( AOB \) of the circle.

### Given Data:
- Radius of the circle \( r = 5 \) cm.
- Angle \( \theta = 2.447 \) radians.
- Area of triangle \( AOB = 8 \) cm².

### Step 1: Area of the Sector \( AOB \)
The area of the sector \( AOB \) is given by the formula:
\[
\text{Area of Sector} = \frac{1}{2} r^2 \theta
\]
Substituting the given values:
\[
\text{Area of Sector} = \frac{1}{2} \times 5^2 \times 2.447 = \frac{1}{2} \times 25 \times 2.447 = 30.5875 \, \text{cm}^2
\]

### Step 2: Area of the Segment
The area of the segment is obtained by subtracting the area of the triangle from the area of the sector:
\[
\text{Area of Segment} = \text{Area of Sector} - \text{Area of Triangle}
\]
Substituting the values:
\[
\text{Area of Segment} = 30.5875 \, \text{cm}^2 - 8 \, \text{cm}^2 = 22.5875 \, \text{cm}^2
\]

### Step 3: Rounding to 3 Significant Figures
Rounding the area of the segment to 3 significant figures:
\[
\text{Area of Segment} \approx 22.6 \, \text{cm}^2
\]

### Final Answer:
The area of the segment is approximately \( 22.6 \, \text{cm}^2 \).

Would you like any further details or explanations?

Here are some related questions for further practice:
1. How would the area change if the radius were doubled?
2. What is the perimeter of the segment?
3. How would you find the length of the chord \( AB \)?
4. What if \( \theta \) were in degrees instead of radians?
5. Can you derive the formula for the area of the segment from first principles?

**Tip:** When working with areas involving sectors and segments, always double-check the units and make sure to convert angles to radians if necessary.

Learn how to calculate the area of segment AOB in a circle with a radius of 5 cm and an angle of 2.447 radians. Includes step-by-step calculation of the area of the sector and subtraction of the area of the triangle. Ideal for Grades 10-12 students studying circle geometry.

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