What I don’t understand is that how would we able to move the vertical circles to a somewhat diagonal position thereby increasing D(for the optimum fit) if the circles X=W/2 ie the circles are perfectly fit

@jaskaran_1

OK. Here is my try without pictures.

Recall that we are now in the case when

- two vertically aligned circles of diameter
**D = W/2**are at the left of rectangle, - lateral surface is cut from the right of rectangle,
- we have a vertical gap between circles and lateral surface,
- the side
**π * D**of lateral surface is horizontal one.

Since we have a vertical gap, let’s draw a vertical line in the middle of this gap,

separating two rectangles (“left” and “right”) from the whole piece of steel.

Next we move bottom circle to the right until it touches this line.

Now we can thicken both circles a bit, preserving them to be touching the sides of “left” rectangle.

Let the new diameter be **D + eps**.

At the same time we can increase horizontal side of lateral surface keeping it the “right” rectangle

and if **eps** is small enough we can make the new side to be **π * (D + eps)**.

Hence we get a correct pieces for the cylinder of diameter **D + eps** by this small perturbation.

Thanks a lot for your help!I was missing the vertical gap(the wasted steel) which instead of taking in the centre(b/w circles and lateral surface) I had taken on the right side of the lateral surface.