A spherical box is the solid defined by $\rho_0\le \rho \le \rho_1,\ \theta_0\le \theta \le \theta_1,\ \phi_0\le \phi\le \phi_1$ in spherical coordinates. It looks like
Picture credit: KhanAcademy.
Is there an elementary way to compute the volume of this spherical box? When I used integrals, I have found that the volume is
$$\frac13 (\rho_1^3-\rho_0^3)(\theta_1-\theta_0)(\cos(\phi_1)-\cos(\phi_0)).$$
Assuming $\phi$ to be the polar angle, a constant value of it corresponds to a circular cone with apex at the origin, which intercepts a spherical cap of surface area $$ A\left( {r,\phi } \right) = 2\pi r^{\,2} \left( {1 - \cos \phi } \right) $$ which gives a ratio wrt the total surface of the sphere of $$ \omega \left( \phi \right) = {{A\left( {r,\phi } \right)} \over {4\pi r^{\,2} }} = {1 \over 2}\left( {1 - \cos \phi } \right) $$ and the ratio comprised between $\phi_0$ and $\phi_1$ will be $$ \omega \left( {\phi _{\,0} ,\phi _{\,1} } \right) = - {1 \over 2}\left( {\cos \phi _{\,1} - \cos \phi _{\,0} } \right) $$
The dihedral angle $\theta_1 - \theta_0$ will then clearly intercept the fraction $$ \omega \left( {\phi _{\,0} ,\phi _{\,1} ,\theta _{\,0} ,\theta _{\,1} } \right) = {{\left( {\theta _{\,1} - \theta _{\,0} } \right)} \over {2\pi }}\omega \left( {\phi _{\,0} ,\phi _{\,1} } \right) = - {{\left( {\theta _{\,1} - \theta _{\,0} } \right)} \over {4\pi }}\left( {\cos \phi _{\,1} - \cos \phi _{\,0} } \right) $$ which in turn corresponds to the fraction of the volume intercepted between the two spheres $$ \eqalign{ & V = \omega \left( {\phi _{\,0} ,\phi _{\,1} ,\theta _{\,0} ,\theta _{\,1} } \right){{4\pi } \over 3}\left( {R^{\,3} - r^{\,3} } \right) = \cr & = - \left( {R^{\,3} - r^{\,3} } \right){1 \over 3}\left( {\theta _{\,1} - \theta _{\,0} } \right)\left( {\cos \phi _{\,1} - \cos \phi _{\,0} } \right) \cr} $$
