Cynthia Lanius

Geometry Online Introduction Distances Midpoints

School-Bus Geometry


You may want to print this square grid paper to use in this exercise.

a city map


Think of a city's streets as a huge square grid. No city is really laid out in exact squares. There are curves, diagonals, dead-ends, and one-way streets. Our school-bus geometry is what mathematicians call an idealized model. We're considering city streets as if they were a perfect square grid, but we know they're not exactly.



Put an X on all the points on the grid that are 4 blocks away from Point A. Remember this is school-bus geometry, and you have to stay on the streets, no cutting across corners.

The definition of a circle is all the points in a plane a given distance from a given point. You just found all the points 4 units from Point A. So could we say this is what a circle looks like in our school-bus geometry? Wow, that's a circle!


one point on a square grid

See the School-Bus Geometry circle You'll have to hit back on your browser to return to this page.

a Euclidean circle on a square grid


Algebra Connection: While we're talking about circles, let's go back to Euclidean Geometry and look at our familiar circles. The equation of a circle centered at the origin in Euclidean geometry is
x2 + y2 = r2.

a Euclidean circle on a square grid
Can you see from the second figure why that is? (Think Pythagorus.)

Challenge Questions:

  1. The equation above doesn't give us our circle in our new geometry. (Confirm this with some points.) What is the equation for our School-Bus Geometry circle?

  2. If the graph of x2 + y2 = r2 is not a circle in this geometry, what would its graph look like?


Geometry Online Introduction Distances Midpoints

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