To determine the area of a sector of a circle, we need to use the formula for the area of a sector, which is given by:
[tex]\[ A = \frac{1}{2} r^2 \theta \][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the circle,
- [tex]\( \theta \)[/tex] is the central angle of the sector in radians,
- [tex]\( A \)[/tex] is the area of the sector.
Given the parameters from the problem:
- The radius [tex]\( r = \frac{1}{2} \)[/tex],
- The central angle [tex]\( \theta = \frac{7 \pi}{6} \)[/tex].
Now let's substitute these values into the formula to find the area of the sector.
First, we calculate [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = \left( \frac{1}{2} \right)^2 = \frac{1}{4} \][/tex]
Next, we substitute the given values into the area formula:
[tex]\[ A = \frac{1}{2} \times \frac{1}{4} \times \frac{7 \pi}{6} \][/tex]
Now we perform the multiplication:
[tex]\[ A = \frac{1}{2} \times \frac{1}{4} \times \frac{7 \pi}{6} = \frac{1}{8} \times \frac{7 \pi}{6} = \frac{7 \pi}{48} \][/tex]
To get the numeric value of the area, we evaluate [tex]\( \frac{7 \pi}{48} \)[/tex]:
Using [tex]\( \pi \approx 3.141592653589793 \)[/tex], we have:
[tex]\[ A \approx \frac{7 \times 3.141592653589793}{48} \approx \frac{21.991148575128544}{48} \approx 0.4581489286485115 \][/tex]
Therefore, the area of the sector is approximately:
[tex]\[ 0.4581489286485115 \][/tex]
