Головні діагоналі куба

Space Diagonals

Suppose we had a cube, $ABCDEFGH$. We want to define some diagonals for this cube, but where do we start?

Well, we can define some diagonals of the cube as the diagonals of the faces of the cube! We can call these face diagonals.

But that's not it. Let's try finding a diagonal that doesn't lie on any of the faces or sides. We can accomplish this by taking two corners opposite from eachother! This gives a diagonal through the interior of the cube that isn't on a face. We call this a space diagonal.

Using the pythagorean theorem, we can see that the length of a space diagonal of a cube is: $$s \sqrt 3$$

Extending this to a rectangular prism of width, length, height, of length $a,b,c$, respectively. We recognize that all space diagonals are equivalent so without loss of generality, we first find the face diagonal of the face $a \times b$, which, by the Pythagorean Theorem, is $\sqrt{a^{2}+b^{2}}$. Afterwards, the space diagonal is calculated using that face diagonal and side c, resulting in the length $\sqrt{a^{2}+b^{2}+c^{2}}$.