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

To explore the packing of circles with a diameter of 24 inches in two different tiling arrangements — square and hexagonal grids — we can compare the efficiency of these two packings in terms of how well they utilize the available area.

### 1. **Square Packing**:
In a square packing arrangement, the circles are placed in a grid where each circle's center is aligned with the vertices of a square.

- **Area of the circle**:  
  The area of each circle with diameter \( d = 24 \) inches can be computed using the formula:  
  \[
  A_{\text{circle}} = \pi r^2 = \pi \left( \frac{d}{2} \right)^2 = \pi (12)^2 = 144\pi \approx 452.39 \, \text{square inches}
  \]
  
- **Area of each square**:
  The square’s side length is equal to the circle’s diameter (24 inches), so the area of one square is:  
  \[
  A_{\text{square}} = d^2 = 24^2 = 576 \, \text{square inches}
  \]

- **Packing efficiency in a square grid**:  
  The efficiency \( E_{\text{square}} \) of the square packing is the ratio of the area of the circle to the area of the square:  
  \[
  E_{\text{square}} = \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{144\pi}{576} \approx 0.785
  \]
  This means that about 78.5% of the area is filled with circles, while the rest is unused space.

### 2. **Hexagonal Packing**:
In hexagonal packing, the circles are arranged so that each circle is surrounded by six others, creating a tighter configuration than the square grid.

- **Area of the hexagon enclosing one circle**:
  The side length of the hexagon is equal to the radius of the circle, \( r = 12 \) inches. The area of a regular hexagon can be calculated using the formula:
  \[
  A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} s^2 = \frac{3\sqrt{3}}{2} (12)^2 = \frac{3\sqrt{3}}{2} \times 144 \approx 374.12 \, \text{square inches}
  \]

- **Packing efficiency in a hexagonal grid**:  
  The efficiency \( E_{\text{hexagon}} \) of the hexagonal packing is the ratio of the area of the circle to the area of the hexagon:
  \[
  E_{\text{hexagon}} = \frac{A_{\text{circle}}}{A_{\text{hexagon}}} = \frac{144\pi}{374.12} \approx 0.815
  \]
  Thus, about 81.5% of the area is filled with circles, making this packing arrangement more efficient than the square grid.

### Conclusion:
- The **hexagonal packing** is more efficient, with about **81.5% area utilization**, compared to the **square packing**, which has about **78.5% area utilization**.
- The hexagonal pattern allows for tighter packing of circles, making it a better option if the goal is to maximize the number of circles that can fit in a given area.

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Explore the efficiency of packing circles with a 24-inch diameter in square and hexagonal grids. This problem demonstrates key geometric concepts, including the calculation of areas and packing efficiency. Ideal for high school students in Grades 9-12.

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