If a circle inas radius 3 cm what is the length of a arc subtendud by 9 centril angle of \( \frac{3 \pi}{8} \) rad

To find the length of an arc subtended by a central angle in a circle, we can use the formula: \[ \text{Length of arc} = \frac{\theta}{360} \times 2\pi r \] where: - \( \theta \) is the central angle in degrees, - \( r \) is the radius of the circle. Given: - Radius of the circle, \( r = 3 \) cm, - Central angle, \( \theta = \frac{3\pi}{8} \) radians. First, we need to convert the central angle from radians to degrees. The conversion factor is that \( 1 \) radian is equal to \( \frac{180}{\pi} \) degrees. So, the central angle in degrees is: \[ \theta_{\text{degrees}} = \frac{3\pi}{8} \times \frac{180}{\pi} = \frac{3 \times 180}{8} = 67.5 \text{ degrees} \] Now, we can substitute the values into the formula to find the length of the arc: \[ \text{Length of arc} = \frac{67.5}{360} \times 2\pi \times 3 \] Let's calculate this. Calculate the value by following steps: - step0: Calculate: \(\frac{67.5}{360}\times 2\pi \times 3\) - step1: Divide the terms: \(\frac{3}{16}\times 2\pi \times 3\) - step2: Multiply the terms: \(\frac{9}{8}\pi \) - step3: Multiply: \(\frac{9\pi }{8}\) The length of the arc subtended by a central angle of \( \frac{3\pi}{8} \) radians in a circle with a radius of 3 cm is approximately 3.534 cm.