1.70. Given a square with side $a$. On each side of the square, outside it, a trapezoid is constructed such that the upper bases of these trapezoids and their lateral sides form a regular dodecagon. Calculate its area.
1.70. The desired area $S=12 S_{\triangle A O C}$, where $O A$ and $O C$ are radii of the circle circumscribed around the square and the dodecagon (Fig.

1.56). Since the side of the square is $a$, then $O A=O C=\frac{a}{\sqrt{2}}$. We have

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-446.jpg?height=453&width=485&top_left_y=631&top_left_x=361)

Fig. 1.56

$S_{\triangle A O C}=\frac{1}{2} O A \cdot C D$, where $C D \perp O A$. But $C D=\frac{1}{2} O C$, since

$\angle A O C=\frac{2 \pi}{12}=\frac{\pi}{6}$. Thus,

$$
S_{\triangle A O C}=\frac{1}{2} \cdot \frac{a}{\sqrt{2}} \cdot \frac{1}{2} \cdot \frac{a}{\sqrt{2}}=\frac{a^{2}}{8} \text {, i.e., } S=\frac{3 a^{2}}{2} \text {. }
$$

Answer: $1.5 a^{2}$.