To determine the area of a [tex]\(45^\circ\)[/tex] sector of a circle, follow these steps:
1. Understand the problem: We are given the total area of the circle, which is [tex]\(24 \, \text{m}^2\)[/tex], and we need to find the area of a sector with a central angle of [tex]\(45^\circ\)[/tex].
2. Determine the fraction of the circle represented by the sector: A circle has a total of [tex]\(360^\circ\)[/tex]. To find the fraction of the circle that this [tex]\(45^\circ\)[/tex] sector represents, divide the angle of the sector by the total angle of the circle:
[tex]\[
\text{Fraction of the circle} = \frac{45^\circ}{360^\circ} = \frac{1}{8}
\][/tex]
3. Calculate the area of the sector: Since the area of the sector is a fraction of the total area of the circle, multiply the total area of the circle by the fraction calculated in the previous step:
[tex]\[
\text{Area of the sector} = \text{Total area of the circle} \times \text{Fraction of the circle} = 24 \, \text{m}^2 \times \frac{1}{8} = 3 \, \text{m}^2
\][/tex]
Thus, the area of the [tex]\(45^\circ\)[/tex] sector of the circle is [tex]\(3 \, \text{m}^2\)[/tex].
The correct answer is:
C. [tex]\(3 \, \text{m}^2\)[/tex]
