Three phase power- phase to phase voltage?

Question

I know that the voltage from phase to phase is 380V when line power (phase to neutral) is 220V, this means 220*sqrt(3)=380, but how you can prove this formula?

Answer 1

Prove it graphically. The three phases are 120 electrical degrees apart.

Draw a line with a length of 220.
Draw another line with a length of 220 so the two make a 120°angle between the two.
Measure the length of the third side of the triangle.
You'll find that it is 380.

If you bisect the 120° angle, you'll have two right triangles with angles of 30, 60, and 90. See the proof in the attached link to derive the √3 factor.

Answer 2

This would occur when hooked up in WYE connection. Your 380V would actually be your line voltage. Phase to phase you are measuring between 2 phases 120 degrees out of phase. But by going to your neutral or where all 3 phases meet you can measure 220. The formula would be, as you said:
V(line)=sqr root of 3 X V(phase).