A circle has an area of [tex]$24 \, m^2$[/tex]. What is the area of a [tex]$45^{\circ}$[/tex] sector of this circle?

A. [tex][tex]$6 \, m^2$[/tex][/tex]
B. [tex]$2 \, m^2$[/tex]
C. [tex]$3 \, m^2$[/tex]
D. [tex][tex]$4 \, m^2$[/tex][/tex]



Answer :

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]