Each side of a cube is 5 inches long, how do you find the lengths of a diagonal of the cube?

1 Answer
Jun 5, 2017

The diagonal length of the cube is:
5sqrt3 " inches"53 inches or ~~8.660" inches"8.660 inches

Explanation:

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.

enter image source here

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

sqrt5050

sqrt50 = sqrt2 xx sqrt(5^250=2×52

sqrt50 = sqrt2 xx 550=2×5

sqrt50 = 5sqrt250=52

color(lime)(AB = 5sqrt2AB=52

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=52

BC = 5BC=5 because BCBC is simply an edge of the cube.

(5sqrt2)^2 + 5^2 = CA^2(52)2+52=CA2

Now we can use algebra to find CACA

sqrt50^2 + 5^2 = CA^2502+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^275=CA2

sqrt75 = CA75=CA

sqrt75 ~~ 8.660758.660

75 = 3 xx 5^275=3×52

sqrt75 = sqrt3 xx sqrt(5^275=3×52

sqrt75 = sqrt3 xx 575=3×5

sqrt75 = 5sqrt375=53

color(blue)(CA = sqrt75CA=75

color(blue)(CA = 5sqrt3CA=53

color(blue)(CA ~~8.660CA8.660


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=53

Hope this helped. :)