To pack circles with a diameter of 24 inches into a square and hexagonal arrangement, we need to analyze two common packing methods:

1. Square Packing

In square packing, each circle is placed next to each other in rows and columns, with their centers aligned to form a grid. The circles are arranged in such a way that the distance between the centers of two adjacent circles is equal to the diameter of the circles.

• Center-to-center distance: 24 inches (equal to the diameter).

• Area per circle:
  The area of a single circle is given by $A_{\text{circle}} = \pi r^2$, where $r = 12$ inches (radius of the circle).
  $A_{\text{circle}} = \pi (12)^2 = 144\pi \approx 452.39$ square inches.

• Area per square cell: Each circle fits into a square cell with side length equal to the diameter of the circle.
  The area per square cell is $24 \times 24 = 576$ square inches.

• Packing Efficiency:
  The packing efficiency is the ratio of the area occupied by the circles to the total area of the square cells.
  Efficiency $E_{\text{square}} = \frac{A_{\text{circle}}}{A_{\text{cell}}} = \frac{144\pi}{576} \approx 0.785$ or 78.5%.

2. Hexagonal Packing

Hexagonal packing, also known as the closest packing, is more efficient. In this arrangement, each circle is surrounded by six other circles, forming a hexagon.

• Center-to-center distance: 24 inches.

• Area per circle: Same as above, $A_{\text{circle}} = 144\pi \approx 452.39$ square inches.

• Area per hexagonal cell:
  The area of a hexagonal cell is based on the distance between the centers of the circles. The hexagon can be divided into 6 equilateral triangles.
  The side length of each triangle is 24 inches (the diameter), and the area of one triangle is $\frac{\sqrt{3}}{4} \times (24)^2 \approx 249.43$ square inches.
  The area of the hexagonal cell is $6 \times 249.43 \approx 1496.58$ square inches.

• Packing Efficiency:
  Efficiency $E_{\text{hex}} = \frac{A_{\text{circle}}}{A_{\text{cell}}} = \frac{144\pi}{1496.58} \approx 0.906$ or 90.6%.

Summary:

• Square packing efficiency: 78.5%
• Hexagonal packing efficiency: 90.6%

Thus, hexagonal packing is more efficient than square packing, and will allow for more circles to be packed into the same space.

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Related questions:

1. How do we calculate the maximum number of circles that can fit in a given area using these packing methods?
2. How do the results change if the circles have a different diameter?
3. What real-world applications use hexagonal packing?
4. How does circle packing efficiency vary in 3D shapes like spheres?
5. Can we use any other packing methods to improve efficiency?

Tip:

Hexagonal packing is widely used in nature (e.g., honeycomb structures) because of its optimal efficiency!