Six-sided Box-lid

We can use the prismoid formula, provided that we dissected this solid correctly: one central cube and six "lids". Due to symmetry, if we draw the cross section, we will discover that the cube has side length $r \sqrt { 2 }$ . The volume of each lid is $\frac { h }{ 6 } \left( 0+4{ r }^{ 2 }\left( \frac { 5 }{ 2 } -\sqrt { 2 } \right) +2{ r }^{ 2 } \right) ,$ where $h = r\left(1-\frac{\sqrt 2}{2}\right)$ .

Adding the volumes of the cube with six times the lid yields $(16-8\sqrt{2}){r}^{3}$ or $(2-\sqrt{2}){D}^{3}$ .

I skipped the geometric steps due to the lack of a diagram, but it takes about one page to show.