# 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:

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

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