Block or cuboid

The wall diagonals of the block have sizes of √29cm, √34cm, √13cm. Calculate the surface and volume of the block.

Correct result:

V =  30
S =  62

Solution:

$$\begin{gathered} 29=a^2+b^2 \\ 34=b^2+c^2 \\ 13=a^2+c^2 \\ {a}^2=29-b^2=29-34+c^2=29-34+13-a^2 \\ 2a^2=x^2-y^2+z^2 \\ a=\sqrt{{\displaystyle \frac{29-34+13}{2}}}=2 \\ \dots \\ b=\sqrt{{\displaystyle \frac{29+34-13}{2}}}=5 \\ \dots \\ c=\sqrt{{\displaystyle \frac{-29+34+13}{2}}}=3 \\ V=a\cdot b\cdot c=2\cdot 5\cdot 3=30 \end{gathered}$$

$S=2 \cdot \ (a \cdot \ b+b \cdot \ c+c \cdot \ a)=2 \cdot \ (2 \cdot \ 5+5 \cdot \ 3+3 \cdot \ 2)=62$

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