My math is rusty and I've been struggling with this most afternoon.
If you have a circle or a cylinder supported by a block, as in the picture below, how do you calculate the distance $x$ when the block is increased from $L_0$ to $L_1$?
For example, supposing the circle is a bottle, and there are blocks on both sides, equally elevating the circle such that the center is elevated vertically. Assume $L_0$ to be the height at which the circle rests on the horizon, and $L_1$ is the increased size of the block such that $L = L_0 + L_1$.
How do you express $x$ in terms of $L_1$?
Let the radius of the circle be $r$. If you draw the vertical radius when it is on the small block and the segment from the center to the far corner of the small block you get a $90-45-45$ triangle, so the hypotenuse is $\sqrt 2$ times a leg. That says $r\sqrt 2 =r+L0\sqrt 2$ or $L0=r(1-\frac 1{\sqrt 2})$ but we don't really care.
If you draw a horizontal chord in the circle at the top of the $L1$ block, draw a vertical radius and a radius to the corner of the $L1$ block you get a right triangle. The horizontal leg is $r-L1$, the hypotenuse is $r$, so the vertical leg is $\sqrt{r^2-(r-L1)^2}=\sqrt{2rL1-L1^2}$. The height of the center is $L1+\sqrt{2rL1-L1^2}$ so the height of the bottom is $$x=L1+\sqrt{2rL1-L1^2}-r$$