To find the area of a sector, we can use the formula:
Area of sector = (θ/360°) × πr²
where θ is the central angle in degrees and r is the radius of the circle. Here, we know the radius (r) is 18.
First, we need to find the central angle θ. The arc length (L) of a sector can be calculated using:
L = (θ/360°) × 2πr
Given that the arc length is 6π and the radius is 18, we can set up the equation:
6π = (θ/360°) × 2π × 18
Simplifying this, we get:
6 = (θ/360°) × 36
Now, multiply both sides by 360°:
6 × 360° = θ × 36
Next, simplifying further:
2160° = θ × 36
This gives us:
θ = 2160° / 36 = 60°
Now that we have θ, we can substitute it back into the area formula:
Area = (60/360) × π × (18)²
Calculating further:
Area = (1/6) × π × 324 = 54π
Thus, the area of the sector is 54π square units.