# 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:
$5 \sqrt{3} \text{ inches}$ or $\approx 8.660 \text{ inches}$

#### 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 $A B$ and $B C$ as the two legs, and $C A$ as the hypotenuse.
We can find the diagonal by using the Pythagoras Theorem.

First, we need to find the length of $A B$.

${a}^{2} + {b}^{2} = {c}^{2}$

${s}^{2} + {s}^{2} = {d}^{2}$

${s}^{2} 2 = {d}^{2}$

$s = 5$

${5}^{2} 2 = {d}^{2}$

We can now use algebra to find out the length of the diagonal of the square, which is the shorter diagonal of the cube.

${5}^{2} 2 = {d}^{2}$

$25 \times 2 = A {B}^{2}$

$50 = A {B}^{2}$

$A {B}^{2} = 50$

$A B = \sqrt{50}$

$\sqrt{50}$

sqrt50 = sqrt2 xx sqrt(5^2

$\sqrt{50} = \sqrt{2} \times 5$

$\sqrt{50} = 5 \sqrt{2}$

color(lime)(AB = 5sqrt2

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

${a}^{2} + {b}^{2} = {c}^{2}$

$A {B}^{2} + B {C}^{2} = C {A}^{2}$

$A B = 5 \sqrt{2}$

$B C = 5$ because $B C$ is simply an edge of the cube.

${\left(5 \sqrt{2}\right)}^{2} + {5}^{2} = C {A}^{2}$

Now we can use algebra to find $C A$

${\sqrt{50}}^{2} + {5}^{2} = C {A}^{2}$

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

$50 + {5}^{2} = C {A}^{2}$

$50 + 25 = C {A}^{2}$

$75 = C {A}^{2}$

sqrt75 = sqrt(CA^2

$\sqrt{75} = C A$

$\sqrt{75} \approx 8.660$

$75 = 3 \times {5}^{2}$

sqrt75 = sqrt3 xx sqrt(5^2

$\sqrt{75} = \sqrt{3} \times 5$

$\sqrt{75} = 5 \sqrt{3}$

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:

$d = \sqrt{3} \times a$

$d = \sqrt{3} \times 5$

color(blue)(d = 5sqrt3

Hope this helped. :)