As per the question it is given that,
Diameter of the hemisphere $=14cm$
Thus, the radius of the hemisphere $=7cm$
Total height of the vessel $=13cm=h+r$
Thus,
Inner surface area of the vessel $=2\pi r\left( h+r \right)$
$=2(22/7)(7)(13)$
$=572c{{m}^{2}}$
Hence, the inner surface area of the vessel is $572c{{m}^{2}}$