The area of a sector of a circle of radius 36 cm is 54 `pi` sq cm. Find the length of the corresponding arc of the sector.

Answer

To find the length of the corresponding arc of the sector, we can follow these steps:

### Step 1: Use the formula for the area of a sector
The area \( A \) of a sector of a circle can be calculated using the formula:
\[ A = \frac{\theta}{360^\circ} \times \pi r^2 \]
where \( \theta \) is the angle in degrees and \( r \) is the radius of the circle.

### Step 2: Substitute the known values
We know the area \( A = 54\pi \) cm² and the radius \( r = 36 \) cm. Substituting these values into the formula gives:
\[ 54\pi = \frac{\theta}{360^\circ} \times \pi \times (36)^2 \]

### Step 3: Simplify the equation
We can cancel \( \pi \) from both sides:
\[ 54 = \frac{\theta}{360^\circ} \times 36^2 \]
Calculating \( 36^2 \):
\[ 36^2 = 1296 \]
Now, substituting this back into the equation:
\[ 54 = \frac{\theta}{360^\circ} \times 1296 \]

### Step 4: Solve for \( \theta \)
To isolate \( \theta \), we multiply both sides by \( 360^\circ \):
\[ 54 \times 360^\circ = \theta \times 1296 \]
Calculating \( 54 \times 360 \):
\[ 54 \times 360 = 19440 \]
So we have:
\[ 19440 = \theta \times 1296 \]
Now, divide both sides by \( 1296 \):
\[ \theta = \frac{19440}{1296} \]
Calculating this gives:
\[ \theta = 15^\circ \]

### Step 5: Find the length of the arc
The length \( L \) of the arc of the sector can be calculated using the formula:
\[ L = \frac{\theta}{360^\circ} \times 2\pi r \]
Substituting \( \theta = 15^\circ \) and \( r = 36 \) cm:
\[ L = \frac{15}{360} \times 2\pi \times 36 \]

### Step 6: Simplify the arc length calculation
Calculating \( \frac{15}{360} \):
\[ \frac{15}{360} = \frac{1}{24} \]
Now substituting this back into the formula:
\[ L = \frac{1}{24} \times 2\pi \times 36 \]
Calculating \( 2 \times 36 = 72 \):
\[ L = \frac{72\pi}{24} \]
Simplifying gives:
\[ L = 3\pi \text{ cm} \]

### Final Answer
The length of the corresponding arc of the sector is \( 3\pi \) cm.

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