Why do this
This is a classic, the historical reference to Archimedes is
educational, and the problem should be part the education of
every student of mathematics. To do this problem requires only
very simple geometry and it introduces the idea of approximation
by finding an upper and lower bounds, and refining the
approximation by taking a series of values where, in this case, we
use smaller and smaller edges, or more and more sides for the
polygons. In addition this problem is a valuable pre-calculus
experience as it uses the idea of a limiting process involving
smaller and smaller ' bits'.
First ask everyone to work out the perimeters of the two squares in
the diagram. Then have a class discussion about what this tells us
about how large the length of the circumference of a circle can be
and how small. Discuss the history of this method with reference to
Archimedes and introduce the idea that it is a method for finding
the value of $\ pi$. Pose the problem: "How would you find the
value of $\pi$ if it was not already known?"
Introduce the idea of an upper bound and a lower bound
for pi and raise the question about how we might improve these
bounds to get closer to the value of pi. Then ask the class to
repeat the exercise using circumscribed and inscribed hexagons.and
Suggest your students researchArchimedes method for finding $\pi$
and other methods of approximating $\pi$ on the internet for
themselves. Discuss the difficulties of calculation, in particular
finding square roots, without modern calculating aids and refer to
the problem Archimedes and
Can you find the perimeter of the square (or other regular polygon)
circumscribing the circle?
Can you find the perimeter of the square (or other regular
polygon) inscribed inside the circle?
What can you say about the lengths of the perimeters of these two
polygons and the length of the circumference of the circle?
Knowing the circumference is $2\pi r$ how does this help you find a
lower and an upper bound for $\pi$.
See Archimedes and Numerical