Homeschool Math: Archimedes and the Method of Exhaustion

With The Story of Science: Aristotle Leads the Way as our spine, the kids and I have been working through a very interesting and hands-on history of math and science. We're using the Student's Quest Guide as a starting point for the hands-on activities, and then I add to them or supplement as needed.

That's exactly what we did with this activity on Archimedes, in which the kids were meant to model his method of approximating pi by calculating the perimeters of inscribed and circumscribed regular polygons for a circle.

Note: The Student's Quest Guide attributes the method of exhaustion to Aristotle, but that's a typo. It was Archimedes.

The Quest Guide had the kids working large-scale, with rope and a meter stick. We chose to also use a tape measure and a protractor triangle.

First the kids drew a large circle on the driveway, exactly the way they did for our sound measurement activity (but with chalk, not stomping in the snow), then drew its diameter, then measured a 90-degree angle from that diameter--

--then used that information to circumscribe a square:

To inscribe a square, you can use the diameters that you've already drawn, or, if you want your inscribed square to line up nicely, you can draw the diagonals of the circumscribed square. Find the points where those lines meet the perimeter of the triangle, and make those the vertices of your inscribed square:

To model Archimedes' method of approximating pi, measure the perimeter of both the circumscribed and inscribed squares and average them, and then divide that by the circle's diameter:

That answer is okay, but it's not terribly accurate, is it?

Want to make it more accurate? Use a regular polygon with more sides!

It got a little crazy trying to do this out on the driveway with chalk and a tape measure, so we moved this activity indoors.

For this, you need a compass, protractor, ruler, and plenty of paper.

We repeated the exercise for inscribing and circumscribing a square, and I let the kids eyeball the circumscribed figures, rather than drawing a diameter to cross the middle and then measuring 90 degrees from it:

The measurements were okay, but not a good approximation of pi.

So we inscribed and circumscribed hexagons instead!

To inscribe a hexagon, draw your diameter, then measure 60 and 120 degrees and draw diameters at those angles.

To circumscribe a hexagon, add diameters drawn at 30 and 150 degrees:

I didn't give the children those instructions, and Syd LOATHED having to use trial-and-error. You can see she's got a few incorrect tries on paper there. Eventually, I told her the degree measurements, but Will was able to complete the task independently.

 Even though my circumscribed hexagon is somehow a little wonky, you can see that I got an approximation of 3.1--that's pretty darn good!

Will tried circumscribing and inscribing an octagon, but the answer wasn't anymore accurate, likely because more lines just means more places for human error. She was VERY impressed when I told her that Archimedes had used a 96-sided figure to make these calculations!

If you can visualize that 96-sided figure, you can see how the more sides you have, the more the figure resembles a circle. 

And that's how you can use a much, much, much more time-consuming method to calculate pi!