Some fun with Archimedes and Pi

Last night I was flipping through one of my favorite books – 100 Great Problems of Elementary Mathematis:

Book pic


I came across the section describing Archimedes’s method of calculating pi and thought it would make a fun morning activity with the boys.  Some of the details of the geometry are a little over their heads, but I didn’t want to get too caught up in the details anyway.  Made for a fun morning.

The first part was just kind of silly – finding a way to introduce pi:

From here we moved on to Archimedes’s method.  We drew a hexagon inscribed in a circle and showed how you could use that hexagon to get the simple estimate that pi = 3:

Playing around with the hexagon inside a circle turned out to be fairly straightforward.  Next we moved on to the slightly more difficult problem – discussing a hexagon circumscribed about the circle.  We studied the picture for a while, used the Pythagorean theorem, and eventually found the perimeter of this hexagon, too.

After this, my instincts led me astray initially.  Luckily, though, I caught myself before moving on, and spent a little time asking the kids to see if they could figure out how to improve our estimate:

The next part is probably the most difficult.  Archimedes figured out a really neat relationship the perimeters of certain polygons and used that relation to get better and better approximations to pi.  The derivation of the relation really uses only the Pythagorean theorem, but I didn’t want to get caught up in the details today:

Finally, we moved to the computer to run through some of the approximations.  I’ve been trying to figure out ways to incorporate a little more computer math, so I was really happy to have the opportunity to do that here.   The last step in our little talk about pi was writing a simple program:

The simple program that we wrote to study the formula is easy to share, you can play with it here if you want:

The kids seemed to really hearing about this way to calculate pi.  It was fun having the chance to walk through all of this with them.