A calculator gives us rational approximations of irrational numbers, so we can “see” that . Explain how a circle inscribed in a regular hexagon allows us to “see” that without any decimal approximations.

Source: NCTM Mathematics Teacher 2008

**SOLUTION**

Consider the circle of center inscribed in a regular hexagon of side length . Triangle is an equilateral triangle. Triangle a 30-60-90 degrees triangle with hypotenuse of length . is also the radius of the circle

Method 1: Area of triangle > area of sector

Simplify

Divide both sides of inequality by and multiply by

Substitute the value of into the above inequality

Simplify

Method 2: Area of hexagon > area of circle

Substitute the value of into the above inequality

Simplify

**Answer**: Given in solution