Cutting a corner

Why does strafe-running has the effect that it does? Well, by going forward, you get a velocity of 50 units in the direction that you are facing. By activating strafing and going left or right, you can get a velocity of 50 units to one side. By doing both at the same time, you move forward and to the side simultaneously, taking you along the diagonal of a square. (This is where diagonal-running, the other common term for this technique, has its origin.) Since the diagonal of this square is longer than the side you'd move along by just going forwards, you cover more ground in the same time.

Diagram showing the geometry of
     strafe-runningDiagram key The net result is that you move at 45o to the angle you are facing, but you also move faster. A little geometry shows that the speed up factor you get should be equal to the square-root of two, 1.412... so the percentage increase is (sqrt(2) - 1) * 100%, or approximately 41%.

That's the basic principle. Actually, things are a little more complicated than that, but don't worry about it unless you are interested.