What is the arc length formula?

The arc length formula you are referring to is derived from the formula for the circumference of a circle. The full circumference of a circle is given by (C = 2 \pi r), where (r) is the radius. When we want to find the length of an arc corresponding to a central angle (\theta) in degrees, we need to express the fraction of the circle that the arc represents.

Since a full circle is (360) degrees, the fraction of the circle that the arc represents is (\frac{\theta}{360}). To obtain the length of the arc, we multiply this fraction by the total circumference:

[

\text{length} = \left(\frac{\theta}{360}\right) \times C = \left(\frac{\theta}{360}\right) \times (2 \pi r)

]

This leads us directly to the formula:

[

\text{length} = \left(\frac{\theta}{360}\right) \times 2 \pi r

]

This formula is useful for calculating the length of a specific arc in a circle when the angle subtended by the arc is known. The correct choice represents this relationship accurately.