The area of a sector of a circle of 6 cm radius is `15 pi` sq. cm. Find the measure of the arc and length of the arc corresponding to the sector.

Answer

To solve the problem, we need to find the measure of the angle (θ) and the length of the arc corresponding to the given sector of a circle with a radius of 6 cm and an area of 15π sq. cm.

### Step 1: Use the formula for the area of a sector
The formula for the area of a sector is given by:

\[ \text{Area of sector} = \frac{\theta}{360} \times \pi r^2 \]

Where:
- θ is the angle in degrees,
- r is the radius of the circle.

### Step 2: Substitute the known values into the formula
We know the area of the sector is \(15\pi\) sq. cm and the radius \(r = 6\) cm. Plugging these values into the formula gives:

\[ 15\pi = \frac{\theta}{360} \times \pi \times (6)^2 \]

### Step 3: Simplify the equation
First, simplify \( (6)^2 \):

\[ (6)^2 = 36 \]

Now, substitute this back into the equation:

\[ 15\pi = \frac{\theta}{360} \times \pi \times 36 \]

### Step 4: Cancel π from both sides
We can cancel π from both sides of the equation:

\[ 15 = \frac{\theta}{360} \times 36 \]

### Step 5: Solve for θ
Now, multiply both sides by \(360\):

\[ 15 \times 360 = \theta \times 36 \]

Calculating \(15 \times 360\):

\[ 5400 = \theta \times 36 \]

Now, divide both sides by 36 to find θ:

\[ \theta = \frac{5400}{36} = 150 \text{ degrees} \]

### Step 6: Find the length of the arc
The formula for the length of the arc (L) is given by:

\[ L = \frac{\theta}{360} \times 2\pi r \]

### Step 7: Substitute θ and r into the arc length formula
Now substitute θ = 150 degrees and r = 6 cm into the formula:

\[ L = \frac{150}{360} \times 2\pi \times 6 \]

### Step 8: Simplify the expression
First, simplify \( \frac{150}{360} \):

\[ \frac{150}{360} = \frac{5}{12} \]

Now substitute this back into the equation:

\[ L = \frac{5}{12} \times 2\pi \times 6 \]

Calculating \(2 \times 6 = 12\):

\[ L = \frac{5}{12} \times 12\pi \]

### Step 9: Cancel out the 12
The 12 in the numerator and denominator cancels out:

\[ L = 5\pi \text{ cm} \]

### Final Answers
- The measure of the angle (θ) is **150 degrees**.
- The length of the arc (L) is **5π cm**.

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