To determine the area of a sector defined by an arc length in a circle, we can follow these steps. Let's dive into it step-by-step:
### 1. Given Information: - Radius ([tex]\( r \)[/tex]) of the circle is 6 units. - Arc length ([tex]\( l \)[/tex]) is [tex]\( 4\pi \)[/tex] units.
### 2. Calculate the Circumference of the Circle: The circumference [tex]\( C \)[/tex] of a circle can be calculated using the formula: [tex]\[ C = 2\pi r \][/tex]
Substitute the given radius: [tex]\[ C = 2\pi \times 6 = 12\pi \][/tex]
The circumference of the circle is [tex]\( 12\pi \)[/tex] units.
### 3. Calculate the Fraction of the Circle Represented by the Sector: We need to ascertain the fraction of the entire circle that the sector represents, and this is found using the arc length [tex]\( l \)[/tex] divided by the circumference [tex]\( C \)[/tex]: [tex]\[ \text{Fraction of Circle} = \frac{l}{C} \][/tex]
Substitute the known values: [tex]\[ \text{Fraction of Circle} = \frac{4\pi}{12\pi} = \frac{4}{12} = \frac{1}{3} \][/tex]
The sector represents [tex]\(\frac{1}{3}\)[/tex] of the circle.
### 4. Calculate the Total Area of the Circle: The area [tex]\( A \)[/tex] of a circle can be calculated using the formula: [tex]\[ A = \pi r^2 \][/tex]
Substitute the given radius: [tex]\[ A = \pi \times 6^2 = \pi \times 36 = 36\pi \][/tex]
The total area of the circle is [tex]\( 36\pi \)[/tex] square units.
### 5. Calculate the Area of the Sector: The area of the sector is calculated by multiplying the fraction of the circle by the total area of the circle: [tex]\[ \text{Area of Sector} = \text{Fraction of Circle} \times \text{Total Area of Circle} \][/tex]
Substitute the values we found: [tex]\[ \text{Area of Sector} = \frac{1}{3} \times 36\pi \][/tex]
[tex]\[ \text{Area of Sector} = 12\pi \][/tex]
### Conclusion: Thus, the area of the sector defined by an arc length of [tex]\( 4\pi \)[/tex] in a circle with radius 6 units is [tex]\( 12\pi \)[/tex] square units. This is approximately [tex]\( 37.699 \)[/tex] square units.
