In the solution video it is said that the radius will not be 6 as the diameter 12 will not fit the other side which is only 10 cm. So the radius is taken as 5. But by that same logic the diameter 10 will also not fit the other side which is 8. Hence shouldn’t the radius be 4?
I believe here that we need to find the largest rectangle on the surface of the box so we can create the largest circle. If the sides of the box are 12x10, 12x8, and 10x8 then the largest single face on the box is the 12x10. So the radius has to be 5 because a circle of radius 6 would not fit here. But we don’t need to make the radius 4 because if we put the circle of radius 5 on the 12x10 side, the 8 represents the height of the cylinder.
If we were to choose the side 10x8 to put the circle, the radius would be 4 and the height would be 12 which calculates out to 16pi x 12 which is less than 25pi x 8.