Circle More Efficient Than Hexagon

To compare the efficiency of a circle and a hexagon, we need to consider the ratio of the area to the perimeter for each shape. This ratio gives an indication of how much area is 'enclosed' per unit length of the boundary, which can be thought of as a measure of compactness or efficiency.

For a circle, the area ( A ) and the perimeter (circumference ( C )) are given by:

( A = \pi r^2 ), where ( r ) is the radius.
( C = 2\pi r ).

The efficiency of a circle can be defined as ( \frac{A}{C} = \frac{\pi r^2}{2\pi r} = \frac{r}{2} ).

For a regular hexagon with side length ( s ), the area ( A ) and perimeter ( P ) are given by:

( A = \frac{3\sqrt{3}}{2} s^2 ), as the area of a regular hexagon can be thought of as six equilateral triangles of side length ( s ).
( P = 6s ).

The efficiency of a hexagon can be defined as ( \frac{A}{P} = \frac{3\sqrt{3}s^2}{12s} = \frac{\sqrt{3}s}{4} ).

To compare these efficiencies directly, we need a common basis. Let's consider both shapes having the same perimeter. For a circle, let ( C = 2\pi r ), and for a hexagon, let ( P = 6s ). Equating these perimeters, ( 2\pi r = 6s ) gives ( s = \frac{\pi r}{3} ).

Substituting ( s = \frac{\pi r}{3} ) into the efficiency of a hexagon, we get:

( \frac{\sqrt{3}\left(\frac{\pi r}{3}\right)}{4} = \frac{\pi\sqrt{3}r}{12} ).

Now, we compare ( \frac{r}{2} ) (circle's efficiency) and ( \frac{\pi\sqrt{3}r}{12} ) (hexagon's efficiency).

The circle's efficiency is directly proportional to its radius, while the hexagon's efficiency depends on both the radius of the circumscribed circle and the constant factor ( \frac{\pi\sqrt{3}}{12} ). Given that ( \pi \approx 3.14 ) and ( \sqrt{3} \approx 1.73 ), the constant factor for the hexagon is approximately ( \frac{3.14 \times 1.73}{12} \approx 0.45 ).

This comparison shows that the circle is more efficient in terms of area per unit perimeter length, as ( \frac{r}{2} ) is greater than ( \frac{\pi\sqrt{3}r}{12} ) for any positive value of ( r ). In simpler terms, a circle encloses more area per unit of perimeter than a regular hexagon of the same perimeter, making the circle more compact and efficient in this sense.