Problem E

Pi Hexagons

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 regular hexagon and a circumscribed regular hexagon 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 regular hexagon \(DEBGFA\) and the circumscribed regular hexagon \(HIJKLM\).

The perimeter of hexagon \(DEBGFA\) 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 hexagon \(HIJKLM\) 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 hexagons, determine a lower and an upper bound for \(\pi\).

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

A line drawn from the centre of a circle perpendicular to a tangent line meets the tangent line at the point of tangency.

For a circle with centre \(C\), the centres of both the inscribed and circumscribed regular hexagons will be at \(C\).