Pyramid in cube

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

Result

V =  576 dm3
S =  465.994 dm2

Solution:

a=12 h=12 V=a2⋅ h/3=122⋅ 12/3=576 dm3a=12 \ \\ h=12 \ \\ V=a^2 \cdot \ h/3=12^2 \cdot \ 12/3=576 \ \text{dm}^3
h2=h2+(a/2)2=122+(12/2)2≐6 5≐13.4164 S2=a⋅ h2/2=12⋅ 13.4164/2≐36 5≐80.4984 S=a2+4⋅ S2=122+4⋅ 80.4984≐465.9938≐465.994 dm2h_{2}=\sqrt{ h^2+(a/2)^2 }=\sqrt{ 12^2+(12/2)^2 } \doteq 6 \ \sqrt{ 5 } \doteq 13.4164 \ \\ S_{2}=a \cdot \ h_{2}/2=12 \cdot \ 13.4164/2 \doteq 36 \ \sqrt{ 5 } \doteq 80.4984 \ \\ S=a^2+4 \cdot \ S_{2}=12^2+4 \cdot \ 80.4984 \doteq 465.9938 \doteq 465.994 \ \text{dm}^2

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