Cube diagonals

The cube has a wall area of 81 cm square. Calculate the length of its edge, wall, and body diagonal.

Result

a =  9 cm
u1 =  12.728 cm
u2 =  15.588 cm

Solution:

S1=81 cm2  a=S1=81=9 cmS_{1}=81 \ \text{cm}^2 \ \\ \ \\ a=\sqrt{ S_{1} }=\sqrt{ 81 }=9 \ \text{cm}
u12=a2+a2=2 a2 u1=2⋅ a=2⋅ 9≐9 2≐12.7279≐12.728 cmu_{1}^2=a^2 + a^2=2 \ a^2 \ \\ u_{1}=\sqrt{ 2 } \cdot \ a=\sqrt{ 2 } \cdot \ 9 \doteq 9 \ \sqrt{ 2 } \doteq 12.7279 \doteq 12.728 \ \text{cm}
u22=a2+a2+a2=3 a2 u2=3⋅ a=3⋅ 9≐9 3≐15.5885≐15.588 cmu_{2}^2=a^2 + a^2 + a^2=3 \ a^2 \ \\ u_{2}=\sqrt{ 3 } \cdot \ a=\sqrt{ 3 } \cdot \ 9 \doteq 9 \ \sqrt{ 3 } \doteq 15.5885 \doteq 15.588 \ \text{cm}

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