h = R - x/2;
a = Sqrt[R^2 - (x/2)^2];
x = R*Sqrt[2];
V = (π/3)*h*(3*a^2 - h^2)

Expand[%]

Factor[%]

N[%]

0.4337634281895244` (R^3)

%/(4*π/3)/
        R^3*1/(R*Sqrt[2]/2)

\(\(\(0.14644660940672627`\/R\)\(
\)\)\)