4.29. A regular $n$-sided polygon is inscribed in a circle of radius $R$, the area of which is equal to $3 R^{2}$. Find $n$.
4.29. We have $\angle A O B=\frac{360^{\circ}}{n}$ and, therefore, $S_{\triangle A O B}=\frac{R^{2}}{2} \sin \frac{360^{\circ}}{n}$ (Fig. 4.26). Thus, we obtain the equation $\frac{R^{2}}{2} \sin \frac{360^{\circ}}{n}=\frac{3 R^{2}}{n}$, which holds

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-586.jpg?height=337&width=387&top_left_y=766&top_left_x=764)

Fig. 4.26 for $n=12$.

Answer: $\boldsymbol{n}=12$.