10.133. A tangent of length $2a$ is drawn from a point $M$, which is at a distance $a$ from the circle, to this circle. Find the area of a regular hexagon inscribed in the circle.
## Solution.

Draw the radius $O A$ to the point of tangency (Fig. 10.120) and denote the radius of the circle by $r$. Then in $\triangle O A M$ we have $(2 a)^{2}+r^{2}=(a+r)^{2}$ or $4 a^{2}+r^{2}=a^{2}+2 a r+r^{2}$, from which $r=\frac{3 a}{2}$. Thus,

$$
\begin{aligned}
& S=\frac{6 r^{2} \sqrt{3}}{4}=\frac{27 a^{2} \sqrt{3}}{8} \\
& \text { Answer: } \frac{27 a^{2} \sqrt{3}}{8}
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-619.jpg?height=438&width=470&top_left_y=79&top_left_x=187)

Fig. 10.121

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-619.jpg?height=432&width=274&top_left_y=80&top_left_x=845)

Fig. 10.122