Question

(PYTHAGOREAN THEOREM)
A diagonal of a cube goes from one of the cube's top corners to the opposite corner of the base of the cube. Find the length to a diagonal d in a cube that has an edge of length 10 meters.

Answer 1

First draw a picture (see attached).

You want to find the diagonal of the cube, length AB.
The right triangle formed is trianagle ABD.
AD = 10m

DB will be the hypotenuse of triangle BCD.
$(BC)^2 + (CD)^2 = (DB)^2 \\ (10)^2 + (10)^2 = (DB)^2 \\ DB = \sqrt{200}= 10 \sqrt{2}$.

If we know the length of AD and DB, we can find AB.
$(AD)^2+(BC)^2 = (AB)^2 \\ (10)^2 + (10 \sqrt{2})^2 = (AB)^2 \\ AB = \sqrt{300} = 10 \sqrt{3}$

In fact, the diagonal of any cube is √3 times the side length of the cube.
Let s be the side length (as in AD or CD in the attached) and
h be the hypotenuse of the base, (DB in the attached) and
d be the diagonal of the cube (AB in the attached).

The hypotenuse of the base will be:$s^2+s^2 = h^2$.

The cube's diagonal will be:
$h^2+s^2 = d^2$.

Substituting $s^2+s^2$ as $h^2$, you have
$s^2+s^2+s^2 = d^2 \\ 3s^2 = d^2 \\ d = \sqrt{3s^2} = s\sqrt{3}$