To pack circles of diameter 24 inches using square and hexagonal arrangements, we can compare the efficiency of the two packing methods in terms of the packing density, which is the ratio of the area occupied by the circles to the total area available in the arrangement.

1. Square Packing

In square packing, the circles are arranged in rows and columns. For each circle, its center is aligned with a grid point of the square.

• Side length of each square cell: 24 inches (equal to the circle's diameter)
• Area of each square cell: $24^2 = 576$ square inches
• Area of each circle: $\pi \times \left(\frac{24}{2}\right)^2 = 452.39$ square inches
• Packing density: $\frac{\text{Area of circle}}{\text{Area of square cell}} = \frac{452.39}{576} \approx 0.785$

So, the packing density for square packing is approximately 78.5%.

2. Hexagonal Packing

In hexagonal packing, each circle is surrounded by six other circles in a hexagonal arrangement. This is known to be the most efficient packing method in terms of density.

• Side length of each hexagon: 24 inches
• Area of each hexagonal cell: The area of a hexagon with side length $r = 12$ inches (radius of the circle) is given by: $\text{Area of hexagon} = \frac{3\sqrt{3}}{2} \times r^2 = \frac{3\sqrt{3}}{2} \times 12^2 \approx 374.12 \text{ square inches}$
• Packing density: The packing density in hexagonal packing is given by: $\frac{\text{Area of circle}}{\text{Area of hexagonal cell}} = \frac{452.39}{374.12} \approx 0.9069$

So, the packing density for hexagonal packing is approximately 90.7%.

Conclusion

• Square packing: 78.5% efficiency
• Hexagonal packing: 90.7% efficiency

Hexagonal packing is more efficient and should be preferred if you want to maximize the number of circles packed in a given area.

Would you like further details or have any questions?

Related Questions:

1. How do you calculate the number of circles that fit in a given area with square packing?
2. How do you calculate the number of circles that fit in a given area with hexagonal packing?
3. How does the packing density change with different circle sizes?
4. What are the real-world applications of circle packing?
5. How does circle packing apply to 3D spaces (e.g., sphere packing)?

Tip:

For large-scale packing, hexagonal arrangements save significant space and are widely used in manufacturing and material science for efficient packing.