. Pack the circles with radius equal to 14cm using square and hexagon.
2. Pack the circles with diameter equal to 24 inches using square

### 1. **Pack the circles with radius 14 cm using square and hexagon:**

#### 1.1 Square Packing:
When packing circles in a square pattern, each circle is arranged in a grid-like format. The distance between the centers of two adjacent circles is equal to the diameter of a circle, which is \(2 \times 14 = 28\) cm.

- **Area of each circle**: 
  \[
  A_{\text{circle}} = \pi r^2 = \pi (14)^2 = 615.75 \text{ cm}^2
  \]

- **Area of each square cell**: 
  Since the circles fit into the square grid, the side of each square cell is equal to the diameter of the circle:
  \[
  A_{\text{square}} = (28)^2 = 784 \text{ cm}^2
  \]

- **Packing efficiency (square packing)**:
  \[
  \text{Packing efficiency} = \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{615.75}{784} \approx 0.785 \text{ or } 78.5\%
  \]
This means that 78.5% of the area is covered by circles, while the remaining 21.5% is wasted space.

#### 1.2 Hexagonal Packing:
Hexagonal packing is more efficient than square packing because each circle touches six others, reducing the wasted space.

- **Packing efficiency (hexagonal packing)**: The hexagonal packing has a well-known efficiency of approximately 90.7%. Therefore:
  \[
  \text{Packing efficiency} \approx 90.7\%
  \]

### 2. **Pack the circles with diameter equal to 24 inches using square pattern:**

In this case, the diameter of each circle is 24 inches, and we are using a square packing method.

- **Area of each circle**: 
  \[
  A_{\text{circle}} = \pi r^2 = \pi (12)^2 = 452.39 \text{ in}^2
  \]
  (Here, the radius is half of the diameter: \( \frac{24}{2} = 12\) inches).

- **Area of each square cell**:
  Since the circles fit into the square grid, the side of each square cell is equal to the diameter of the circle:
  \[
  A_{\text{square}} = (24)^2 = 576 \text{ in}^2
  \]

- **Packing efficiency (square packing)**:
  \[
  \text{Packing efficiency} = \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{452.39}{576} \approx 0.785 \text{ or } 78.5\%
  \]
The packing efficiency is again approximately 78.5%, the same as in the first case since the pattern and relative dimensions are the same.

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### Would you like any further details or explanations on these results?

#### 5 Related Questions:
1. How does hexagonal packing work geometrically?
2. What are some practical applications of circle packing in industry?
3. Can we calculate the packing efficiency for other shapes, such as triangles?
4. How does the efficiency of circle packing change with varying circle sizes?
5. What is the best way to pack circles in a rectangular space?

#### Tip:
Hexagonal packing is often the most efficient method for packing circles and is found in nature, such as in honeycomb structures, due to its minimal wasted space.

Pack the circles with radius equal to 14 cm using square and hexagon. Pack the circles with diameter equal to 24 inches using square.

Learn how to pack circles with a radius of 14 cm using both square and hexagon patterns. Explore packing efficiency calculations and compare them. Also, discover how to pack circles with a diameter of 24 inches in a square pattern. Suitable for Grades 10-12 geometry students.

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