10.179. A rhombus, in which the side is equal to the smaller diagonal (Fig. 10.148), is equal in area to a circle of radius $R$. Determine the side of the rhombus.
Solution.

From the condition $\triangle A B D$ is equilateral, therefore, $\angle B A D=60^{\circ}$.

Area of the rhombus: $S_{\mathrm{p}}=x^{2} \sin 60^{\circ}=\frac{\sqrt{3}}{2} x^{2}=\pi R^{2} \Rightarrow x=\sqrt{\frac{2 \pi R^{2}}{\sqrt{3}}}=R \sqrt{\frac{2 \pi}{\sqrt{3}}}$.

Answer: $\quad R \sqrt{\frac{2 \pi}{\sqrt{3}}}$.