Problem D

Pi Squares

Pi Day is an annual celebration of the mathematical constant \(\pi\). Pi Day is observed on March \(14\), since \(3\), \(1\), and \(4\) are the first three significant digits of \(\pi\).

Archimedes determined lower bounds for \(\pi\) by finding the perimeters of regular polygons inscribed in a circle with diameter of length \(1\). (An inscribed polygon of a circle has all of its vertices on the circle.) He also determined upper bounds for \(\pi\) by finding the perimeters of regular polygons circumscribed in a circle with diameter of length \(1\). (A circumscribed polygon of a circle has all sides tangent to the circle. That is, each side of the polygon touches the circle in one spot.)

In this problem, we will determine a lower bound for \(\pi\) and an upper bound for \(\pi\) by considering an inscribed square and a circumscribed square in a circle of diameter \(1\).

Consider a circle with centre \(C\) and diameter \(1\). Since the circle has diameter \(1\), it has circumference equal to \(\pi\). Now consider the inscribed square \(ABDE\) and the circumscribed square \(FGHJ\).

The perimeter of square \(ABDE\) will be less than the circumference of the circle, \(\pi\), and will thus give us a lower bound for the value of \(\pi\). The perimeter of square \(FGHJ\) will be greater than the circumference of the circle, \(\pi\), and will thus give us an upper bound for the value of \(\pi\).

Using these squares, determine a lower bound and an upper bound for \(\pi\).

Note: For this problem, you may want to use the following known results about circles:

For a circle with centre \(C\), the diagonals of an inscribed square meet at \(90\degree\) at \(C\).

For a circle with centre \(C\), the diagonals of a circumscribed square meet at \(90\degree\) at \(C\).

If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency.