To derive the formula for the arc length of a sector, we start with a few fundamental concepts:
1. The circumference [tex]\(C\)[/tex] of a circle is given by the formula [tex]\(C = 2 \pi r\)[/tex], where [tex]\(r\)[/tex] is the radius of the circle.
Now, if we divide the circle into equally sized central angles, each with a measure of [tex]\(n^\circ\)[/tex], we need to determine the number of such sectors in the circle.
2. Because the total number of degrees in a circle is [tex]\(360^\circ\)[/tex], the number of sectors formed, each with a central angle of [tex]\(n^\circ\)[/tex], is given by:
[tex]\[ \frac{360^\circ}{n^\circ} \][/tex]
Each sector corresponds to an arc of the circle. To find the length of each arc (the arc length [tex]\(L\)[/tex]), we need to divide the entire circumference [tex]\(2 \pi r\)[/tex] by the number of sectors:
[tex]\[ L = \frac{2 \pi r}{\frac{360^\circ}{n^\circ}} = 2 \pi r \cdot \frac{n^\circ}{360^\circ} \][/tex]
So, the arc length [tex]\(L\)[/tex] of a sector with a central angle of [tex]\(n^\circ\)[/tex] is:
[tex]\[ L = 2 \pi r \cdot \frac{n}{360} \][/tex]
This is the final expression for the arc length.
Therefore, the correct answer is:
D. [tex]\(2 \pi r \cdot \frac{360}{n}\)[/tex]
