Pressure force exerted by one half of the sphere into his other half

Question

lets say we have some sphere under some pressure and I want to find what is the force pressing one-half of the sphere toward the opposite half?,lets take one piece of area then the pressure force exerted in this area is:

$$p\Delta A$$

and the horizontal component is: $$p\Delta A cos\theta$$

book says the following:

A cos θ is the projection of the area ΔA on a vertical plane, i.e., it is the fraction of the area facing to the right, in silhouette. For the complete half sphere, the net area facing to the right is πR^2 and the net force toward the left is therefore pπR^2

I dont understand that statment.why the area is not onehalf of that, can someome offer me an intuitive explanation of what happen, why the net force to the left is pπR^2?

Answer 1

The force is being exerted onto the flat, circular face of the hemisphere, which has area $\pi R^2$. Since that face is flat, the force at each point on that face points to the left, so that the net force on that face of the hemisphere is $p\pi R^2$ to the left. We know that the hemisphere as a whole is at rest, so the net force on the curved bit of the hemisphere must be $p\pi R^2$ to the right.