# Space diagonal

The space diagonal of a cube is 129.91 mm. Find the lateral area, surface area and the volume of the cube.

Result

L =  5625 mm2
S =  33750 mm2
V =  421875 mm3

#### Solution:

$d=129.91 \ \text{mm} \ \\ \ \\ d^2=a^2+a^2+a^2 \ \\ d^2=3a^2 \ \\ \ \\ d=\sqrt{ 3 } a \ \\ \ \\ a=[ { d/ \sqrt{ 3 } } ]=[ { 129.91/ \sqrt{ 3 } } ]=75 \ \text{mm} \ \\ \ \\ L=a^2=75^2=5625 \ \text{mm}^2$
$S=6 \cdot \ L=6 \cdot \ 5625=33750 \ \text{mm}^2$
$V=a^3=75^3=421875 \ \text{mm}^3$

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