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

### 1. Packing Circles with Radius of 14 cm in Square and Hexagonal Grids

- **Square Grid Packing:**

In square packing, circles are arranged in rows and columns where each circle touches its neighbors along the x and y axes.

- The side length of the square cell needed to pack one circle will be equal to the diameter of the circle.
- The diameter of the circle is \( 2r = 2 \times 14 \, \text{cm} = 28 \, \text{cm} \).
- Therefore, each circle fits into a square cell of side 28 cm.

- **Hexagonal Grid Packing:**

Hexagonal packing is more efficient than square packing. In this arrangement, circles are packed in a way that every circle touches six surrounding circles.

- The distance between the centers of two adjacent circles in a hexagonal grid is equal to the circle's diameter.
- However, the height of each "hexagonal row" is shorter than in a square packing, allowing for a denser arrangement. The vertical distance between two rows of circles is \( \sqrt{3}r \), which gives us:
  \[
  \text{Vertical distance} = \sqrt{3} \times 14 \, \text{cm} \approx 24.25 \, \text{cm}.
  \]
  
Therefore, the packing efficiency is greater in the hexagonal packing.

### 2. Packing Circles with Diameter of 24 Inches in Square and Hexagonal Grids

- **Square Grid Packing:**

The side length of the square cell for each circle will be equal to the circle’s diameter:
\[
\text{Side length of square} = 24 \, \text{inches}.
\]
Each circle fits into a square with a side length of 24 inches.

- **Hexagonal Grid Packing:**

In hexagonal packing, the distance between the centers of two adjacent circles is still 24 inches, but the vertical distance between rows is shorter, calculated as:
\[
\text{Vertical distance} = \sqrt{3} \times 12 \, \text{inches} \approx 20.78 \, \text{inches}.
\]

### Summary of Packing Characteristics:
- **Square Packing Efficiency:** Space is used less efficiently, with some gaps between the circles.
- **Hexagonal Packing Efficiency:** More efficient packing arrangement with less wasted space between the circles.

#### Do you want more details on packing efficiency or further calculations?

### Related Questions:
1. How does the packing density differ between square and hexagonal grids?
2. How many circles can fit in a square with a side of 100 cm using each packing method?
3. What is the ratio of unused space in square and hexagonal packings?
4. How does circle size impact the efficiency of hexagonal packing?
5. What is the real-world application of these packing methods?

### Tip: Hexagonal packing is more efficient and often used in nature, such as in honeycombs!

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

Explore how to pack circles with a radius of 14 cm and a diameter of 24 inches using square and hexagonal grids. This problem covers key geometry concepts, including circle diameter, square grid side length, and hexagonal packing efficiency. Suitable for high school students in Grades 9-12.

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