10.085. A circle of radius $R$ is inscribed in an isosceles trapezoid. The upper base of the trapezoid is half the height of the trapezoid. Find the area of the trapezoid.
Solution.

$C D=y ; E D=x$ (Fig. 10.82). We solve the system $\left\{\begin{array}{l}2 y=(R+2 x)+R, \\ x^{2}+4 R^{2}=y^{2} .\end{array}\right.$ From this, we get $x=\frac{3}{2} R$, then $A D=R+2 x=R+3 R=4 R$. The required area $S=\frac{1}{2}(A D+B C) \cdot E C=\frac{1}{2}(R+4 R) \cdot 2 R=5 R^{2}$.

Answer: $5 R^{2}$.