Using the formula derived from the Pythagorean Theorem we can find the distance between any two points.  In my geometry class, we talked about just changing the idea of path of travel.  The question was posed, "What if you were only allowed to travel along the grid lines?"  So, the segment AB would no longer be allowed.  This is where the grid is treated like streets, and you would travel like a Taxi-Cab would travel.  When going from point A to point B, like a car, one would have to first travel to point P and then on to point B.  Now our distance is measured by the Distance from A to P plus the distance from P to B.
 

One of the things that we explored in the Geometry class were different figures.  The first thing that we did was sketch all the points a given Distance[Taxi] from a point.  Click here and try the activity.  Given point A, sketch all points with a Distance[Taxi]= 3.

Did you get something similar to mine?...

At first this looks much like our traditional square.  But, if you go back and read how we created this figure, you'll see that we actually created a Circle[Taxi].  (All points equal distance from a given point).  So, what we would typically deam a square is actually a circle.  And just by simply changing the definition of distance between two points.  Hmmmm...