1.35. Calculate the ratio of the areas of a square, an equilateral triangle, and a regular hexagon inscribed in the same circle.

## Group B
1.35. Let $R$ be the radius of the circle. Then the side of the inscribed regular triangle is $a_{3}=R \sqrt{3}$ and $S_{3}=\frac{a_{3}^{2} \sqrt{3}}{4}=\frac{3 R^{2} \sqrt{3}}{4}$. Next, the side of the square is $a_{4}=R \sqrt{2}$ and $S_{4}=a_{4}^{2}=2 R^{2}$, and finally, the side of the inscribed regular hexagon is $a_{6}=R$ and $S_{6}=\frac{6 R^{2} \sqrt{3}}{4}=\frac{3 R^{2} \sqrt{3}}{2}$. Therefore, $S_{4}: S_{3}: S_{6}=8: 3 \sqrt{3}: 6 \sqrt{3}$.

Answer: $8: 3 \sqrt{3}: 6 \sqrt{3}$.

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-429.jpg?height=463&width=431&top_left_y=1182&top_left_x=46)

Fig. 1.30