Question

find the length of arc ab. (there is a circle with center, radius 6 cm, and the central angle for arc ab is 120°)

Explanation:

Step1: Recall the arc length formula

The formula for the length of an arc is \( L=\frac{\theta}{360^\circ}\times2\pi r \), where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle.

Step2: Identify the values of \( \theta \) and \( r \)

From the diagram, the radius \( r = 6\space\text{cm} \) and the central angle \( \theta=120^\circ \).

Step3: Substitute the values into the formula

Substitute \( \theta = 120^\circ \) and \( r = 6\space\text{cm} \) into the arc length formula:
\[

$$\begin{align*} L&=\frac{120^\circ}{360^\circ}\times2\pi\times6\\ &=\frac{1}{3}\times12\pi\\ & = 4\pi\\ &\approx4\times3.14\\ &=12.56\space\text{cm} \end{align*}$$

\]

Answer:

The length of arc \( AB \) is \( 4\pi\space\text{cm} \) (or approximately \( 12.56\space\text{cm} \)).