Packing circles inside shapes like squares and hexagons involves arranging the circles in such a way that they occupy the maximum possible area within the shape, minimizing wasted space. Here’s how this works for both the square and hexagon:

1. Packing Circles in a Square:

• Square Area Calculation:

  • The side length of the square needed to accommodate the circles depends on the arrangement. The most efficient packing arrangement for circles in a square is a hexagonal close-packing, even though the enclosing shape is a square.
  • For a square, the circles will be arranged in rows and columns. The closest packing arrangement is when each circle in one row fits into the gap between two circles in the adjacent rows.

• Calculation:

  • If each circle has a radius $r = 14$ cm, the diameter $d$ is $2r = 28$ cm.
  • In the hexagonal arrangement within a square:
    • The side of the square $L$ needed to accommodate $n$ circles along the diagonal is approximately: $L \approx 2r \times \sqrt{n} \times \sqrt{\frac{2}{\sqrt{3}}}$
    • However, you could also calculate based on direct packing rows and columns.

2. Packing Circles in a Hexagon:

• Hexagon Area Calculation:

  • The most efficient way to pack circles inside a hexagon is to align them in a way that the centers of the circles correspond to the vertices and the midpoints of the hexagon's sides.

• Calculation:

  • For a hexagon with side length $a$, the number of circles that can fit inside depends on the relationship between the side length of the hexagon and the diameter of the circles.
  • If the hexagon has a side length equal to the diameter of the circle, the number of circles $N$ that fit inside the hexagon can be derived from: $N \approx \frac{2A}{\sqrt{3}r^2}$ where $A$ is the area of the hexagon.

Summary:

• Square: In a square, the circles are usually packed in rows and columns. Hexagonal packing within a square maximizes efficiency, but the square’s edges may lead to partial circles at the boundaries.
• Hexagon: In a hexagon, the circles can be packed more efficiently due to the shape’s alignment with the hexagonal packing structure.

Would you like me to calculate the exact number of circles that can fit inside a specific square or hexagon of your choice? Or provide a detailed explanation with diagrams?

Relative Questions:

1. How does the packing density change when transitioning from a square to a hexagon?
2. What are the mathematical implications of packing circles in a hexagonal grid inside a square?
3. Can we derive a formula to calculate the number of circles in a given polygonal area?
4. How do edge effects influence the total number of circles that can fit inside irregular shapes?
5. What are practical applications of circle packing in real-world scenarios?

Tip: For maximum packing efficiency, use hexagonal packing whenever possible—it’s the most space-efficient way to arrange circles.