What is the arc length formula?

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Multiple Choice

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.

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.

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