We know that Archimedes discovered a primitive
approach to determine pi before calculus was
invented. He used simple trigonometry to approximate
the value of pi. Let's start with a circle
of diameter 1, which makes the circumference
of the circle equal to pi. Now draw a square
around it. We can clearly see the circumference
of the circle is less than the perimeter of
the square, which equals 4. Next let's inscribe
another square inside the circle. Clearly
the circumference of this circle is greater
than the perimeter of the square. We can figure
out the length one side like so: we know that
the hypotenuse of this triangle is 1 and we
do not know the opposite or adjacent side.
Since they are equal we can solve for any
of them. Sine 45 equals the opposite over
1, the opposite side is equal to root 2 over
2, opposite equals adjacent, and we know that
the perimeter of the square is equal to 4,
so 4 x root 2 over 2 gives us approximately
2.828. In this case we know that pi is less
than 4 but greater than 2.828. You can get
a better approximation by adding more and
more sides to your circumscribed and inscribed
polygons and working out their perimeters.
Archimedes went as far as a 96 sided polygon,
and got the approximation of pi to be less
than 3 1/7 but greater than 3 10/71.
