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Let's solve the problem step by step.

### Problem Breakdown:
- **Given:**
  - Sector \( OAB \) of a circle with center \( O \) and radius \( 5 \) cm.
  - The length of the arc \( AB \) is \( 6 \) cm.

- **Tasks:**
  1. Find the area of the sector \( OAB \).
  2. Find the area of the shaded region.

### Step 1: Find the Area of Sector \( OAB \)

#### 1a. Determine the angle \( \theta \) of the sector:

The length of the arc \( AB \) is given by the formula:
\[
\text{Arc length} = r \theta
\]
where \( r = 5 \) cm and the arc length is \( 6 \) cm. Thus:
\[
6 = 5 \theta
\]

Solve for \( \theta \):
\[
\theta = \frac{6}{5} = 1.2 \text{ radians}
\]

#### 1b. Calculate the area of the sector \( OAB \):

The area of a sector is given by:
\[
\text{Area of sector } OAB = \frac{1}{2} r^2 \theta
\]
Substituting the values:
\[
\text{Area of sector } OAB = \frac{1}{2} \times 5^2 \times 1.2 = \frac{25 \times 1.2}{2} = 15 \text{ cm}^2
\]

So, the area of the sector \( OAB \) is \( 15 \) cm².

### Step 2: Find the Area of the Shaded Region

The shaded region is the area of the sector minus the area of triangle \( OAB \).

#### 2a. Area of Triangle \( OAB \):

To find the area of the triangle \( OAB \), we can use the following formula:
\[
\text{Area of triangle } OAB = \frac{1}{2} r^2 \sin(\theta)
\]
where \( r = 5 \) cm and \( \theta = 1.2 \) radians.

\[
\text{Area of triangle } OAB = \frac{1}{2} \times 5^2 \times \sin(1.2)
\]
Using a calculator to find \( \sin(1.2) \approx 0.932 \):
\[
\text{Area of triangle } OAB = \frac{1}{2} \times 25 \times 0.932 = 11.65 \text{ cm}^2
\]

#### 2b. Area of Shaded Region:

Now subtract the area of triangle \( OAB \) from the area of sector \( OAB \):
\[
\text{Area of shaded region} = \text{Area of sector } OAB - \text{Area of triangle } OAB
\]
\[
\text{Area of shaded region} = 15 - 11.65 = 3.35 \text{ cm}^2
\]

### Final Answer:
- **Area of the sector \( OAB \)**: \( 15 \) cm²
- **Area of the shaded region**: \( 3.35 \) cm²

---

Would you like further details or have any questions? Here are some related questions to consider:

1. What would be the area of the sector if the radius was doubled?
2. How does the area of the shaded region change if the angle \( \theta \) is halved?
3. Can you find the perimeter of the sector \( OAB \)?
4. How would the area of the shaded region be calculated if \( OAB \) were a segment instead of a sector?
5. What would happen if the length of the arc \( AB \) was given as a function of \( \theta \)?

**Tip:** When dealing with sectors and triangles in circle-related problems, always double-check if the angles are in radians or degrees before calculating areas.

Learn how to calculate the area of sector OAB and the shaded region with a detailed step-by-step solution. This problem involves circle geometry, sector area calculation, and triangle area formulas. Suitable for high school students.

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