Try different radius sizes. See how the area to perimeter ratios are affected. What shape has the highest area/perimeter ratio? Now, in a grid of shapes, where the perimeter is made of a costly material, which grid of shapes holds the most area? A grid of circle containers, a grid of square containers or a grid of hexagon containers (like a beehive)? See area/side calculations (a/s in grid).
Shows regular polygons' area & perimeter. Things to notice: For any given shape, as the area and radius increase, the ratio of area to perimeter increases. As shapes approximate a circle, the area/perimeter ratio increases. But a grid of circles isn't as efficient as a grid of hexagons, in terms of area/perimeter material. Why? Imagine a grid of circle containers and imagine a grid of hexagon containers, each made of wax. Circles can't share material of their sides. A grid of circles leave lots of space between the circles. In contrast, hexagons can share each of their sides with another hexagon in all the inner portions of the grid. So, that's a way to picture why the hexagonal structure of beehives is the most efficient grid of containers. For a radius of 100: square area/perimeter = 35 hexagon area/perimeter = 43 circle area to perimeter = 50 But squares share each side with another square, so for a grid of squares, area/perimeter approximates 70. Even a grid of squares is doing better than a grid of circles. Now, remember hexagons' area/perimeter is greater than that of squares. A grid of hexagons share sides with another hexagon. so a grid of hexagons area/perimeter approximates 86
