Pyramid in cube

In a cube with an edge 12 dm long, we have an inscribed pyramid with the apex at the center of the cube's upper wall. Calculate the volume and surface area of the pyramid.

Correct result:

V =  576 dm³
S =  465.9938 dm²

Solution:

$$\begin{gathered} a=12 \\ h=12 \\ V=a^2\cdot h\mathrm{/}3=12^2\cdot 12\mathrm{/}3=576{\text{dm}}^3 \end{gathered}$$

$$\begin{gathered} h_2=\sqrt{h^2+(a\mathrm{/}2{)}^2}=\sqrt{12^2+(12\mathrm{/}2{)}^2}=6\sqrt{5}\doteq 13.4164 \\ {S}_2=a\cdot h_2\mathrm{/}2=12\cdot 13.4164\mathrm{/}2=36\sqrt{5}\doteq 80.4984 \\ S=a^2+4\cdot S_2=12^2+4\cdot 80.4984=465.9938{\text{dm}}^2 \end{gathered}$$

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