Delicious Candy Solids!

Use the prismoidal formula to figure out the volume of this weird prism.

$V=\dfrac { 1 }{ 6 } h\left( { A }_{ 1 }+4{ A }_{ 2 }+{ A }_{ 3 } \right)$

where $h$ is height

$h=\sqrt { { \left( \dfrac { 1 }{ 2 } \sqrt { 3 } \right) }^{ 2 }-{ \left( \dfrac { 1 }{ 2 } \left( \sqrt { 2 } -1 \right) \right) }^{ 2 } }$

and ${A}_{1}, {A}_{2}, {A}_{3}$ are the cross section areas at bottom, mid-section, and top

${A}_{1}={A}_{3}=1$

and octagon mid-section area

${A}_{2}=2(1+\sqrt{2})\dfrac{1}{{2}^{2}}$

so that the volume works out to

$V=\dfrac{1}{3}\sqrt{4+3\sqrt{2}}=0.957...$

so the unit cube has the barely larger volume