A regular 12-sided polygon is inscribed in a circle of radius 3 cm. We need to find the side length of the polygon. Step 1: Formula for the Side Length of an Inscribed Polygon For a regular \( n \)-sided polygon inscribed in a circle of radius \( R \), the length of each side \( s \) is given by: \[ s = 2R \sin \frac{\pi}{n} \] where: - \( R = 3 \) cm (given radius) - \( n = 12 \) (since it's a 12-sided polygon) Step 2: Substituting Values \[ s = 2(3) \sin \frac{\pi}{12} \] \[ s = 6 \sin \frac{\pi}{12} \] Step 3: Computing \( \sin \frac{\pi}{12} \) Using the identity: \[ \sin 15^\circ = \sin \frac{\pi}{12} = \frac{\sqrt{6} - \sqrt{2}}{4} \] \[ s = 6 \times \frac{\sqrt{6} - \sqrt{2}}{4} \] \[ s = \frac{6(\sqrt{6} - \sqrt{2})}{4} \] \[ s = \frac{3(\sqrt{6} - \sqrt{2})}{2} \] Multiplying by 6: \[ s = 18 - 9\sqrt{3} \] Final Answer: \[ 18 - 9\sqrt{3} \text{ cm} \]