Geometry question

Here's a nice geometry question a student recently asked me about:

The surface area of a cube is 150 sq. cm. What is the length of its diagonal in cm?

The one measurement that's always useful when we're looking at cubes is the length of the side. How do we figure that out here?

Well, there are six sides to a cube, they're all the same size and they're all squares. So each of them has an area of 150÷6 = 25 square centimetres. If the area of a square is 25, the length of the side is √25 = 5cm.

Now comes the tricky bit. When I'm doing 3-d geometry problems, I find it helps to think about a rectangular room with four rectangular walls. In this case, we're trying to find the distance from one corner on the floor to the opposite corner at the top of the wall, and I imagine a bit of string running from one to the other.

Straight away, I can see a right-angled triangle - one side runs diagonally across the floor and the other up the wall, and the string makes the hypotenuse.

We know how big the wall is - it's 5cm, like all of the other sides of the cube. However, we don't know how far it is across the floor. To find that, we need to consult with everyone's favourite bean-eating ancient Greek mathematician, Pythagoras.

The line across the floor is also the hypotenuse of a right-angled triangle - again, its shorter sides are both 5cm long. Pythagoras says the hypotenuse squared is equal to one of the short sides squared plus the other short side squared: H² = 5² + 5² = 50. So the line across the floor is √50 cm long - a smidge over 7cm.

Now we have to work out the longer diagonal. The line on the floor is now one of the shorter sides, along with the wall. So D² = (√50)² + 5² = 50 + 25 = 75. The main diagonal is √75 cm - around 8.66cm.

Bonus question: why is this the same as 5√3?