Hard math contest trigonometry type problem

Above, $O$ is the centre of the sphere, $x$ is its radius. Writing the area of trapezoid $ABCD$ as a sum of three triangles, we can easily solve for $x$ to get:

$$ x=\frac{1200\sqrt{3}}{54}=38.490017\ldots $$

No trig or calculus needed, only geometry.