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" or ~~8.660" inches"

Explanation:

To find the diagonal of the cube, there are 2 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 A to B and then we can construct a triangle with AB and BC as the two legs, and CA as the hypotenuse.
We can find the diagonal by using the Pythagoras Theorem.

First, we need to find the length of AB.

enter image source here

a^2 + b^2 = c^2

s^2 + s^2 = d^2

s^2 2 = d^2

s = 5

5^2 2 = d^2

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^2

25 xx 2 = AB^2

50 = AB^2

AB^2 = 50

AB = sqrt50

sqrt50

sqrt50 = sqrt2 xx sqrt(5^2

sqrt50 = sqrt2 xx 5

sqrt50 = 5sqrt2

color(lime)(AB = 5sqrt2

Now we can construct the triangle and find the overall longer diagonal of the cube.

a^2 + b^2 = c^2

AB^2 + BC^2 = CA^2

AB = 5sqrt2

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

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

Now we can use algebra to find CA

sqrt50^2 + 5^2 = CA^2

The square root and square of 50 cancel each other out.

50 + 5^2 = CA^2

50 + 25 = CA^2

75 = CA^2

sqrt75 = sqrt(CA^2

sqrt75 = CA

sqrt75 ~~ 8.660

75 = 3 xx 5^2

sqrt75 = sqrt3 xx sqrt(5^2

sqrt75 = sqrt3 xx 5

sqrt75 = 5sqrt3

color(blue)(CA = sqrt75

color(blue)(CA = 5sqrt3

color(blue)(CA ~~8.660


And if you wanted to do it the other way, the formula is:

d = sqrt3 xx a

d = sqrt3 xx 5

color(blue)(d = 5sqrt3

Hope this helped. :)