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

### Packing Circles of Diameter 24 Inches Using Square and Hexagonal Arrangements

The task is to arrange circles of diameter 24 inches in both a square and a hexagonal packing pattern. Let's break down the details of each packing method and how efficiently the circles will fit in each.

### 1. **Square Packing**
In square packing, circles are arranged in a grid where each circle is directly aligned with its neighbors. This arrangement is straightforward but less efficient in using space.

#### - **Packing Density (Square Packing):**
The packing density for a square arrangement is given by the formula:
\[
\text{Density} = \frac{\pi}{4} \approx 0.785
\]
This means that about 78.5% of the space in the square grid is occupied by circles, while the rest is wasted.

#### - **Arrangement:**
- The centers of the circles are placed on a square grid with side lengths equal to the circle's diameter (24 inches).
- Each row and column will be spaced 24 inches apart.

### 2. **Hexagonal Packing**
Hexagonal packing is more efficient, as each circle is surrounded by six others, fitting them together more closely.

#### - **Packing Density (Hexagonal Packing):**
The packing density for a hexagonal arrangement is given by the formula:
\[
\text{Density} = \frac{\pi}{2\sqrt{3}} \approx 0.906
\]
This means that approximately 90.6% of the space is occupied by circles, making this arrangement more space-efficient than square packing.

#### - **Arrangement:**
- The centers of the circles are placed on a hexagonal grid where the distance between the centers of adjacent circles is equal to their diameter (24 inches).
- In this arrangement, each circle touches six neighboring circles.

### **Comparison and Summary**
1. **Square Packing:**
   - Circles are spaced evenly in a grid.
   - Packing density: 78.5%.
   - Less efficient in using space.

2. **Hexagonal Packing:**
   - Circles are arranged in a honeycomb-like structure.
   - Packing density: 90.6%.
   - More efficient in using space.

For a given area, hexagonal packing allows you to fit more circles than square packing, making it the preferred method when trying to maximize the number of circles in a given space.

Would you like further details on how to calculate the number of circles that can fit in a particular area using these packing methods?

### Additional Questions
1. How would the number of circles change if the packing area had a specific size?
2. Can you explore how packing density changes if the circle size is adjusted?
3. How does the difference in packing efficiency impact practical applications, like material usage?
4. What are some real-life examples where hexagonal packing is used?
5. Could you compare other shapes' packing efficiency in similar ways?

### Tip
When maximizing space usage with circular objects, hexagonal packing is almost always the best choice due to its high packing density!

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

Explore how to pack circles with a 24-inch diameter using square and hexagonal arrangements. Learn the differences in packing density for square (78.5%) and hexagonal (90.6%) grids, and understand how these geometric patterns affect space efficiency. Suitable for high school geometry students.

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