Pack the circles with radius equal to 14 cm using square and hexagon

Packing circles inside shapes like a square or a hexagon is a classical problem in geometry, focusing on how many circles can fit within a given space. Given that the circles have a radius of 14 cm, let’s discuss the packing efficiency in both a square and a hexagon.

### 1. **Packing Circles in a Square**
When packing circles in a square, the most efficient method is a hexagonal or triangular lattice arrangement. However, for simplicity, consider both the rectangular grid and the hexagonal packing.

- **Square Packing (Rectangular Grid)**: In this case, the circles are arranged in rows and columns. Each circle takes up a square area of \( (2r)^2 \), where \( r = 14 \, \text{cm} \), so each circle fits inside a square of side 28 cm.
- **Hexagonal Packing in Square**: This is more efficient than a rectangular grid. The circles are packed in rows, with each alternate row offset. This arrangement reduces wasted space.

#### Square Packing Formulas:
- The area of a single circle is \( A_{\text{circle}} = \pi \times r^2 \), where \( r = 14 \, \text{cm} \).
- The area of the square is \( A_{\text{square}} = \text{side}^2 \), where the side depends on the total number of circles packed.

### 2. **Packing Circles in a Hexagon**
Hexagonal packing within a hexagonal area is even more efficient. The most optimal packing method aligns the circles in a hexagonal arrangement where each circle is surrounded by six others.

#### Hexagon Packing Formulas:
- The radius of each circle is 14 cm, and in a hexagonal packing, the centers of the circles form an equilateral triangle grid.
- The area of a regular hexagon with circumradius \( r = 14 \, \text{cm} \) is given by:
  \[
  A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} \times r^2
  \]
- The packing density in a hexagonal arrangement is around 0.9069, meaning about 90.69% of the hexagonal area will be covered by circles.

Would you like to see specific examples or calculations for a given square or hexagon size?

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### Related Questions:
1. What is the efficiency difference between square and hexagonal packing for larger numbers of circles?
2. How does the size of the square or hexagon affect the number of circles that can fit?
3. Can we fit more circles if we allow overlapping arrangements or different radii?
4. How does the perimeter length of the square or hexagon affect the packing efficiency?
5. What is the optimal arrangement for circles in other polygonal shapes?

### Tip:
Hexagonal packing is generally more space-efficient than square packing due to its higher packing density (around 91% compared to 78.5% for square packing).

Explore how to efficiently pack circles with a 14 cm radius inside a square and a hexagon. This problem covers key geometric concepts, including area calculation and packing density, suitable for students in Grades 10-12.

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