Problem of the Week 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:

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

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

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