A cone of radius R and height H is hanging inside a liquid of density $\rho$ by means of a string as shown in the figure. The force, due to the liquid acting on the slant surface of the cone is

The Buoyancy force is the weight of the liquid that is been displaced by the cone. 

The Net force on the either sides of the cone the resultant buoyancy force.

Force at the bottom surface=$\left(\mathrm{\rho gH}\right)\times {\mathrm{\pi R}}^2$

And the net buoyancy force = volume of cone x density of fluid x acceleration due to gravity

$=\left(\frac{1}{3}{\mathrm{\pi R}}^2\mathrm{H}\right)\times \mathrm{\rho g}$

The force acting on the slant surface of the cone $=\left({\mathrm{\rho gH\pi R}}^2\right)-\left(\frac{1}{3}{\mathrm{\pi R}}^2\mathrm{H\rho g}\right)=\frac{2}{3}{\mathrm{\pi \rho gHR}}^2$