The area of a sector of a circle of radius 36 cm is `72 tc cm^(2)`.The length ofthe corresponding arc of the sector is

Answer

To find the length of the corresponding arc of a sector of a circle given the area of the sector, we can follow these steps:

### Step 1: Understand the formula for the area of a sector
The area \( A \) of a sector of a circle is given by the formula:
\[ A = \frac{1}{2} r^2 \theta \]
where \( r \) is the radius of the circle and \( \theta \) is the angle in radians.

### Step 2: Substitute the known values
We know the area of the sector is \( 72\pi \) cm² and the radius \( r = 36 \) cm. We can substitute these values into the area formula:
\[ 72\pi = \frac{1}{2} \times 36^2 \times \theta \]

### Step 3: Calculate \( 36^2 \)
First, calculate \( 36^2 \):
\[ 36^2 = 1296 \]

### Step 4: Substitute \( 36^2 \) back into the equation
Now substitute \( 1296 \) back into the equation:
\[ 72\pi = \frac{1}{2} \times 1296 \times \theta \]

### Step 5: Simplify the equation
Multiply both sides by 2 to eliminate the fraction:
\[ 144\pi = 1296\theta \]

### Step 6: Solve for \( \theta \)
Now, divide both sides by \( 1296 \):
\[ \theta = \frac{144\pi}{1296} \]
Simplifying this gives:
\[ \theta = \frac{\pi}{9} \text{ radians} \]

### Step 7: Use the formula for the length of the arc
The length \( L \) of the arc of the sector is given by:
\[ L = r \theta \]
Substituting the values we have:
\[ L = 36 \times \frac{\pi}{9} \]

### Step 8: Calculate the length of the arc
Now, simplify:
\[ L = 4\pi \text{ cm} \]

### Final Answer
The length of the corresponding arc of the sector is \( 4\pi \) cm.

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