To find the diagonal of the cube, there are 22 methods, using a formula or by using the Pythagorus Theorem. So that this explanation is more interesting - and so that you're not just putting numbers into a formula - I am going to tell you how to do the Pythagorus theorem way.
To do this, we need to find the length of the diagonal of the face, in this diagram, from AA to BB and then we can construct a triangle with ABAB and BCBC as the two legs, and CACA as the hypotenuse.
We can find the diagonal by using the Pythagoras Theorem.
First, we need to find the length of ABAB.
a^2 + b^2 = c^2a2+b2=c2
s^2 + s^2 = d^2s2+s2=d2
s^2 2 = d^2s22=d2
s = 5s=5
5^2 2 = d^2522=d2
We can now use algebra to find out the length of the diagonal of the square, which is the shorter diagonal of the cube.
5^2 2 = d^2522=d2
25 xx 2 = AB^225×2=AB2
50 = AB^250=AB2
AB^2 = 50AB2=50
AB = sqrt50AB=√50
sqrt50√50
sqrt50 = sqrt2 xx sqrt(5^2√50=√2×√52
sqrt50 = sqrt2 xx 5√50=√2×5
sqrt50 = 5sqrt2√50=5√2
color(lime)(AB = 5sqrt2AB=5√2
Now we can construct the triangle and find the overall longer diagonal of the cube.
a^2 + b^2 = c^2a2+b2=c2
AB^2 + BC^2 = CA^2AB2+BC2=CA2
AB = 5sqrt2AB=5√2
BC = 5BC=5 because BCBC is simply an edge of the cube.
(5sqrt2)^2 + 5^2 = CA^2(5√2)2+52=CA2
Now we can use algebra to find CACA
sqrt50^2 + 5^2 = CA^2√502+52=CA2
The square root and square of 5050 cancel each other out.
50 + 5^2 = CA^250+52=CA2
50 + 25 = CA^250+25=CA2
75 = CA^275=CA2
sqrt75 = sqrt(CA^2√75=√CA2
sqrt75 = CA√75=CA
sqrt75 ~~ 8.660√75≈8.660
75 = 3 xx 5^275=3×52
sqrt75 = sqrt3 xx sqrt(5^2√75=√3×√52
sqrt75 = sqrt3 xx 5√75=√3×5
sqrt75 = 5sqrt3√75=5√3
color(blue)(CA = sqrt75CA=√75
color(blue)(CA = 5sqrt3CA=5√3
color(blue)(CA ~~8.660CA≈8.660
And if you wanted to do it the other way, the formula is:
d = sqrt3 xx ad=√3×a
d = sqrt3 xx 5d=√3×5
color(blue)(d = 5sqrt3d=5√3
Hope this helped. :)