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The image presents two problems regarding circle packing inside squares and hexagons.

### Problem 1: 
Pack circles with a **radius of 14 cm** using square and hexagonal packing.

1. **For Square Packing:**
   In square packing, circles are arranged in a grid where each circle touches four other circles at a 90-degree angle. The area occupied by one circle and its surrounding square grid is:
   \[
   A_{\text{square}} = (2r)^2 = (2 \times 14)^2 = 28^2 = 784 \text{ cm}^2
   \]
   The area of one circle is:
   \[
   A_{\text{circle}} = \pi r^2 = \pi \times 14^2 \approx 615.75 \text{ cm}^2
   \]
   The packing efficiency is the ratio of the area of the circle to the area of the square:
   \[
   \eta_{\text{square}} = \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{615.75}{784} \approx 0.7854 \, (78.54\%)
   \]

2. **For Hexagonal Packing:**
   In hexagonal packing, each circle is surrounded by six others in a honeycomb pattern. The area of a hexagon surrounding each circle can be found using the formula for the area of a regular hexagon inscribed around a circle of radius \( r \):
   \[
   A_{\text{hexagon}} = 6 \times \frac{\sqrt{3}}{4} \times (2r)^2 = 6 \times \frac{\sqrt{3}}{4} \times (28)^2 = 6 \times \frac{\sqrt{3}}{4} \times 784 \approx 1357.17 \text{ cm}^2
   \]
   The packing efficiency in a hexagonal grid is higher:
   \[
   \eta_{\text{hexagon}} = \frac{A_{\text{circle}}}{A_{\text{hexagon}}} = \frac{615.75}{1357.17} \approx 0.9069 \, (90.69\%)
   \]

### Problem 2: 
Pack circles with a **diameter of 24 inches** using square and hexagonal packing.

1. **For Square Packing:**
   The radius is half the diameter, so \( r = 12 \) inches. The area of the square surrounding each circle is:
   \[
   A_{\text{square}} = (2r)^2 = (2 \times 12)^2 = 24^2 = 576 \text{ in}^2
   \]
   The area of one circle is:
   \[
   A_{\text{circle}} = \pi r^2 = \pi \times 12^2 \approx 452.39 \text{ in}^2
   \]
   The packing efficiency is:
   \[
   \eta_{\text{square}} = \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{452.39}{576} \approx 0.7854 \, (78.54\%)
   \]

2. **For Hexagonal Packing:**
   The area of a regular hexagon surrounding a circle of radius 12 inches is:
   \[
   A_{\text{hexagon}} = 6 \times \frac{\sqrt{3}}{4} \times (2r)^2 = 6 \times \frac{\sqrt{3}}{4} \times (24)^2 = 6 \times \frac{\sqrt{3}}{4} \times 576 \approx 998.08 \text{ in}^2
   \]
   The packing efficiency is:
   \[
   \eta_{\text{hexagon}} = \frac{A_{\text{circle}}}{A_{\text{hexagon}}} = \frac{452.39}{998.08} \approx 0.4534 \, (45.34\%)
   \]

Would you like more details on any part of the solution?

### Related Questions:
1. What are other shapes that can be used to pack circles efficiently?
2. How does the radius of circles affect packing efficiency in different patterns?
3. What are real-world applications of circle packing in geometry?
4. How does the packing efficiency change when the container's shape is irregular?
5. How does 3D packing differ from 2D packing in terms of efficiency?

### Tip: 
Hexagonal packing is the most efficient way to pack circles in a plane since it allows the most compact arrangement of circles.

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

Explore how to pack circles with a radius of 14 cm and diameter of 24 inches using square and hexagonal packing methods. Learn about the packing efficiency, area calculations, and the mathematical concepts behind circle packing. Ideal for high school geometry students.

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