In addition to AURA’s reasoning, I’ve come up with another compelling argument for why an x-drive is approximately sqrt2 times faster than a tank drive, given the same gear ratio.
For this example we imagine an x-drive with 4" wheels. (As @Martin said, a similar tank drive with 4" wheels travels approximately 12.566" forward when each wheel makes a full revolution.)
The reasoning is as such:
Imagine first rotating the front right and back left wheels for a full revolution. This diagonal motion moves the robot approximately 8.886" forward (since the robot is travelling diagonally, and therefore to the left the same amount). Imagine next doing the same for the front left and back right wheels. This maneuver also brings the robot approximately 8.886" forward. Both maneuvers, when combined, involve all four wheels rotating one full revolution, with a total forward motion of approximately 17.771". (The shape of the robot’s path approximates a “V” rotated 90 degrees clockwise.)
Now imagine doing the same, but switching diagonal directions after each set of wheels has completed only half a rotation, and continuing until each wheel has made a full revolution. The result is the same, with a forward motion of approximately 17.771", but the robot’s path is closer to a straight line. (The shape of the robot’s path approximates a "W"rotated 90 degrees clockwise.)
We can continue to increase the switching of diagonals until the robot’s path approximates a straight line (i.e. all wheels rotate concurrently.) Still the forward motion will be 17.771", which happens to be sqrt2 times the distance of the same tank drive (12.566").
@Martin if this argument does not compel you, I strongly recommend you build your own x-drive and test your theory out. It is a good exercise and I believe will help you better understand the reasoning behind an x-drive’s speed.