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

The problem of packing circles efficiently inside a square and a hexagon can be approached using two common packing arrangements: **square packing** and **hexagonal packing**. Here's a breakdown for both cases with circles of diameter 24 inches.

### 1. **Packing Circles in a Square (Square Packing Arrangement)**

In a square arrangement, the circles are arranged in a grid, where the center of each circle is aligned both horizontally and vertically with the next circle. This is a simple layout but not the most efficient in terms of space usage.

- The side length of the square (let's call it \( L \)) should accommodate both the number of circles horizontally and vertically.
- For each circle with a diameter of 24 inches, the centers of two adjacent circles are 24 inches apart.

#### Formula for Number of Circles:
If you have a square of side \( L \), the number of circles that can fit along one side is:
\[
n = \frac{L}{24}
\]
Where \( n \) is the number of circles along each side, and \( L \) is the side length of the square.

Thus, the total number of circles you can pack in a square is:
\[
N = n^2 = \left( \frac{L}{24} \right)^2
\]

This gives the total number of circles that can fit in a square packing arrangement.

### 2. **Packing Circles in a Hexagon (Hexagonal Packing Arrangement)**

In a hexagonal arrangement, the circles are packed in a staggered pattern, where each circle is surrounded by six neighboring circles. This is the most efficient packing arrangement in terms of space usage, as it maximizes the density of the circles.

- In this arrangement, the centers of the circles form an equilateral triangle.
- For circles with diameter 24 inches, the distance between the centers of adjacent circles is also 24 inches, but each row is staggered.

#### Formula for Number of Circles:
To calculate the number of circles that can be packed in a hexagonal area, we need to account for the staggered pattern. The packing density of hexagonal packing is known to be approximately 0.9069, meaning 90.69% of the space is occupied by the circles.

If the hexagonal region has an area \( A \), the approximate number of circles you can pack is:
\[
N = \frac{0.9069 \times A}{\pi \times \left(\frac{24}{2}\right)^2}
\]

Here, \( \pi \times \left( \frac{24}{2} \right)^2 \) is the area of each circle.

### Summary:

- **Square Packing:** The number of circles that fit depends on the side length \( L \) of the square, and you can fit \( N = \left( \frac{L}{24} \right)^2 \) circles.
  
- **Hexagonal Packing:** The number of circles depends on the area \( A \) of the hexagon, and the number of circles is approximately:
  \[
  N = \frac{0.9069 \times A}{\pi \times \left(\frac{24}{2}\right)^2}
  \]

These formulas will help you calculate the number of circles for any specific dimensions of a square or hexagonal container.

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Would you like more details on how to calculate the area of the hexagon or a specific example calculation for a given size?

### Follow-up Questions:
1. How do we calculate the side length of a square or hexagon for a given number of circles?
2. What is the difference in space efficiency between square and hexagonal packing?
3. How do we handle non-perfect square or hexagon shapes (e.g., partial circles at the edges)?
4. Can the packing density in hexagonal packing vary with container shape?
5. What are the real-world applications of efficient circle packing?

### Tip:
Hexagonal packing is the most efficient arrangement for packing circles in most cases, as it uses around 90.69% of the available space.

Explore how to efficiently pack circles with a 24-inch diameter in square and hexagonal containers. Learn about the formulas for square and hexagonal packing arrangements, including detailed steps and theorems. Suitable for students in Grades 10-12.

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