To pack circles with a diameter of 24 inches into both square and hexagonal grids, the packing efficiency differs based on the geometry of the layout. Here's a comparison between the two:

1. Square Packing:

In a square packing arrangement, the circles are arranged in rows and columns. The side of the square is equal to the circle's diameter (24 inches in this case). For this layout:

• Area of one circle:
  $A_{circle} = \pi \times \left(\frac{24}{2}\right)^2 = \pi \times 12^2 = 144\pi \, \text{square inches}$
• Area of one square cell:
  $A_{square} = 24 \times 24 = 576 \, \text{square inches}$

The packing density (fraction of area occupied by the circles) is the ratio of the area of the circle to the area of the square: $\eta_{square} = \frac{144\pi}{576} = \frac{\pi}{4} \approx 0.785 \, \text{or} \, 78.5\%$

This means in square packing, about 78.5% of the space is occupied by the circles.

2. Hexagonal Packing:

In hexagonal packing, the circles are arranged such that each circle is surrounded by six others in a hexagonal grid. This is the most efficient way to pack circles.

• Area of one circle:
  $A_{circle} = 144\pi \, \text{square inches}$ (same as before)

• Area of the hexagonal cell:
  The hexagon that contains one circle can be divided into 6 equilateral triangles. The side length of the triangle is equal to the radius of the circle (12 inches).

  The area of one triangle is: $A_{triangle} = \frac{\sqrt{3}}{4} \times 12^2 = 36\sqrt{3}$

  So, the area of the hexagonal cell: $A_{hexagon} = 6 \times 36\sqrt{3} = 216\sqrt{3} \approx 373.18 \, \text{square inches}$

The packing density (fraction of area occupied by the circles) is: $\eta_{hexagon} = \frac{144\pi}{216\sqrt{3}} \approx 0.9069 \, \text{or} \, 90.69\%$

In hexagonal packing, about 90.69% of the space is occupied by the circles.

Conclusion:

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

Thus, hexagonal packing is significantly more efficient than square packing for packing circles.

---

Would you like further details or have any questions?

Here are some related questions to consider:

1. How can I calculate the number of circles that fit in a given area using hexagonal packing?
2. How does packing efficiency change with different circle sizes?
3. What is the mathematical proof for hexagonal packing being the most efficient?
4. How does packing efficiency affect industrial designs (e.g., packaging or storage)?
5. Can packing be optimized with irregular shapes?

Tip: In real-life applications, always account for edge effects when fitting shapes into a finite space!