<h3>What will be the diameter and height of the cylinder obtained from a cube and what are the formulas for the diagonal of a cube and the volume of a cylinder?</h3>

If a cylinder with the biggest possible volume is cut inside the cube, the height of the cylinder and the diameter of the cylinder will be equal to the side length of the cube.
For example, consider the following figure in which the cylinder is cut inside of the cube and since the side length of the cube is $x$ , the diameter and the height of the cylinder are also $x$
If the side length of a cube is $x$ unit, then its diagonal will be $x\sqrt{3}$ unit.
The formula for the volume of a cylinder is $V=\pi r^2h$ , where $r$ is the radius and $h$ is the height of the cylinder. If $d$ is the diameter, then $r=\frac{d}{2}$ .

Now, given that the diagonal of the cube is $24$ cm. So, if the side length of the cube is $x$ cm, then we must have

$x\sqrt{3}=24\\\Longrightarrow x=\frac{24}{\sqrt{3}}\\\Longrightarrow x=8\sqrt{3}$

Thus, the side length of the cube is $8\sqrt{3}$ cm.

So, the height of the cylinder with maximum volume will be $h=8\sqrt{3}$ cm and the diameter will be $d=8\sqrt{3}$ cm i.e. the radius will be $r=\frac{d}{2}=\frac{8\sqrt{3}}{2}=4\sqrt{3}$ cm.