The problem asks us to find the arc length of a circle given the central angle and the radius. The central angle is $225^\circ$ and the radius is $3$ cm.

1. Problem Description

The problem asks us to find the arc length of a circle given the central angle and the radius. The central angle is $225^\circ$ and the radius is $3$ cm.

2. Solution Steps

To find the arc length, we first need to convert the angle from degrees to radians. To convert degrees to radians, we multiply by $\frac{\pi}{180^\circ}$.

$$225^\circ \times \frac{\pi}{180^\circ} = \frac{225\pi}{180} = \frac{5\pi}{4} \text{ radians}$$

The formula for arc length $s$ is given by $s = r\theta$, where $r$ is the radius and $\theta$ is the angle in radians. In this case, $r = 3$ cm and $\theta = \frac{5\pi}{4}$.

$$s = r\theta = 3 \times \frac{5\pi}{4} = \frac{15\pi}{4}$$

Using $\pi \approx 3.14$, we can approximate the arc length:

$$s = \frac{15 \times 3.14}{4} = \frac{47.1}{4} = 11.775 \text{ cm}$$

This is approximately equal to 11.78 cm.

3. Final Answer

1

1. 78 cm