Area of a sector of a circle of radius 16 cm is 256 cm`""^(2)`. Find the length of the corresponding arc of the sector.

Answer

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

### Step 1: Understand the given information
We are given:
- Radius (r) = 16 cm
- Area of the sector = 256 cm²

### Step 2: Use the formula for the area of a sector
The formula for the area of a sector is:
\[ \text{Area} = \frac{r^2 \theta}{2} \]
where \( \theta \) is in radians.

### Step 3: Substitute the known values into the formula
Substituting the values we have into the area formula:
\[ 256 = \frac{16^2 \theta}{2} \]

### Step 4: Calculate \( 16^2 \)
Calculate \( 16^2 \):
\[ 16^2 = 256 \]
So the equation becomes:
\[ 256 = \frac{256 \theta}{2} \]

### Step 5: Simplify the equation
Now simplify the equation:
\[ 256 = 128 \theta \]

### Step 6: Solve for \( \theta \)
To find \( \theta \), divide both sides by 128:
\[ \theta = \frac{256}{128} = 2 \text{ radians} \]

### Step 7: Use the formula for the length of the arc
The formula for the length of the arc (L) of the sector is:
\[ L = r \theta \]
Substituting the values of \( r \) and \( \theta \):
\[ L = 16 \times 2 \]

### Step 8: Calculate the length of the arc
Now calculate the length:
\[ L = 32 \text{ cm} \]

### Final Answer
The length of the corresponding arc of the sector is **32 cm**.

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