VL=Sqrt(3)*Vph reason for sqrt 3

Please draw phasor diagram of 3 phases (the center is the neutral) and calculate the Line to line Voltage

Simple mathematics

http://cr4.globalspec.com/thread/31212/root-3-how-it-comes-3-phase-power-calculations

Pythagoras requires 2 right triangles to complete the logic; Isosceles had it in 1 triangle. Here is a link you can play with, but the basic relationship 1:1:√3 defines the two equal legs and the base.

I applaud the correctness and detail of your answer, but my instinct tells me that simple basics will be better understood in this case.

There was this car once . . . any relation?

Regards, CJM

Think carefully about the triangular relationship between the voltage on each phase, and draw it out, as above, inside a circle that touches all three triangle points on the circle.

If one bisects the equilateral triangle, and applies Pythagoras' Theorem to the right-angled triangles formed, the √3 appears in the outcome. It is the relationship between the centre-to-one-point length and the one-point-to-another-point distance on the triangle, and therefore √3 is the ratio of phase-to-phase voltage divided by the phase-to-neutral voltage, in a 3-phase system.

Look up isosceles triangles and you will note that with 2 equal legs of value "1" and an angle between them of 120⁰, the base will be = square root of 3.

This applies directly to the 120⁰ separation of electrical phases.