# Problem of the Week 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:

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

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