Packing circles in different tessellations (square and hexagonal) is a common problem in geometry that highlights how efficiently space can be used. The circle you’re working with has a diameter of 24 inches, and we’ll explore how circles can be packed using square and hexagonal patterns.

1. Square Packing of Circles:

In a square arrangement, each circle is placed at the vertices of a grid of squares. This leads to gaps between the circles, making the packing less efficient compared to other methods.

• Diameter of each circle: 24 inches
• Area of each circle: $A_{\text{circle}} = \pi \left(\frac{24}{2}\right)^2 = 452.39 \, \text{square inches}$
• Side length of each square: 24 inches (because the diameter of the circle equals the side length of the square)
• Area of each square: $A_{\text{square}} = 24^2 = 576 \, \text{square inches}$
• Packing efficiency: The efficiency is the ratio of the circle's area to the square's area:

$\text{Efficiency}_{\text{square}} = \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{452.39}{576} \approx 0.785$

This means that about 78.5% of the space is used for circles, while the rest is wasted.

2. Hexagonal Packing of Circles:

In a hexagonal packing (or "honeycomb" packing), the circles are arranged so that each circle is surrounded by six others, leading to a more efficient use of space.

• Distance between circle centers: This is still 24 inches, as each circle touches its neighbors at a single point.
• Packing efficiency: For hexagonal packing, the packing efficiency is known to be:

$\text{Efficiency}_{\text{hexagonal}} = \frac{\pi}{2\sqrt{3}} \approx 0.9069$

This shows that about 90.7% of the space is used for circles, making it more efficient than square packing.

Summary:

• Square packing efficiency: ~78.5%
• Hexagonal packing efficiency: ~90.7%

The hexagonal packing is much more efficient for packing circles, utilizing space better than the square packing.

Would you like more details or examples on this? Here are five related questions:

1. What is the maximum number of circles that can fit in a given rectangular space?
2. How can you calculate the packing density for irregularly shaped objects?
3. How do these packing patterns change in 3D (e.g., with spheres)?
4. What are the real-world applications of packing problems?
5. How do packing problems differ when using ellipses or other shapes instead of circles?

Tip: For practical applications, hexagonal packing is often preferred due to its higher efficiency in using available space.