Pack the circles with diameter equal to 24 inches using square and hexagon

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!

Pack the circles with diameter equal to 24 inches using square and hexagon.

Explore how to pack circles with a diameter of 24 inches using square and hexagonal arrangements. Learn about the geometry, packing efficiency, and relevant formulas such as the area of a circle, square packing efficiency, and hexagonal packing efficiency. Understand why hexagonal packing is more efficient, with a 90.6% efficiency rate compared to 78.5% for square packing.

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