how to measure diagonal of hyper-rectangle ?

Question

it's easy to measure it if the rectangle only has 2 dimensions but what if it has multidimension, like 4 or even 10 dimensions. please help me, all genious people out there

Answer 1

Let's take the three dimensional case.

You know that the diagonal of one of the faces has length the square root of the sum of the squares of the two sides of of the face. This is because you can draw a right triangle with the two sides and the diagonal, then use the Pythagorean Theorem.

Well, now look at the face diagonal, the third side, and the real diagonal. They form a right triangle too!

So use the Pythagorean Theorem again and do the algebra, and what do you get? You should find that the diagonal length of the three-dimensional figure is the square root of the sum of the squares of all three sides.

There you have it!

The same technique works for as many dimensions as you like. The diagonal of a lower-dimensional rectangular figure is just a side of a right triangle one dimension higher.