It is given that,
The diameter of the cylinder $=$ the height of the cylinder
$⇒h=2r$, where h – height of the cylinder and r – radius of the cylinder
As we know that,
Volume of a cylinder $=\pi {{r}^{2}}h$
Then, volume of the cylindrical vessel $=\pi {{r}^{2}}2r=2\pi {{r}^{3}}$….. (i)
Now,
Volume of each identical vessel
Diameter $=42cm$, so the radius $21cm$
Height $=21cm$
Then, the volume of two identical vessels $=2\times \pi {{21}^{2}}\times 21$ ….. (ii)
As, the volumes on equation (i) and (ii) are equal
Equating both the equations, we have
$2\pi {{r}^{3}}=2\times \pi {{21}^{2}}\times 21$
${{r}^{3}}={{\left( 21 \right)}^{3}}$
$r=21cm$
So, $d=42cm$
Therefore, the diameter of the cylindrical vessel is $42cm$.