Pack the circle with radius equal to 14 cm using square and hexagon

To pack a circle with a radius of 14 cm using a square and a hexagon, we need to analyze how each of these shapes can be inscribed within the circle. Let's break down the process:

### 1. **Square Inscription**:
A square inscribed in a circle has its diagonal equal to the diameter of the circle. To calculate the side length of the square:

- The diameter of the circle is twice the radius:  
  \[
  \text{Diameter} = 2 \times 14 = 28 \, \text{cm}
  \]
  
- The diagonal of the square is related to the side length \( s \) by the formula:
  \[
  \text{Diagonal} = s\sqrt{2}
  \]
  
- Set the diagonal equal to the diameter of the circle:
  \[
  s\sqrt{2} = 28
  \]
  
- Solving for \( s \):
  \[
  s = \frac{28}{\sqrt{2}} = 28 \times \frac{\sqrt{2}}{2} = 28 \times 0.707 \approx 19.8 \, \text{cm}
  \]
  
Thus, the side length of the square inscribed in the circle is approximately **19.8 cm**.

### 2. **Hexagon Inscription**:
A regular hexagon can also be inscribed in the circle. The relationship between the radius of the circle and the side length \( a \) of the hexagon is simple: the side length of the hexagon is equal to the radius of the circle.

Thus, the side length of the hexagon inscribed in the circle is:
\[
a = 14 \, \text{cm}
\]

### Comparison and Packing:

- **Square**: A square inscribed within the circle will have a larger area coverage than the hexagon but will leave more unused space (corners outside the square but inside the circle).
- **Hexagon**: A hexagon fills the circle more efficiently than the square, as it fits closely to the curvature of the circle with less wasted space.

### Areas:
- The area of the square is:
  \[
  A_{\text{square}} = s^2 = 19.8^2 \approx 392.04 \, \text{cm}^2
  \]
  
- The area of the hexagon is:
  \[
  A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} a^2 = \frac{3\sqrt{3}}{2} (14^2) \approx 509.73 \, \text{cm}^2
  \]

The hexagon covers more area than the square.

Would you like more details on how to compare the efficiencies or practical applications of these packings?

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**Related Questions**:
1. How would the area change if the radius were doubled?
2. How is the packing efficiency calculated for other regular polygons?
3. Can multiple squares or hexagons be packed within one circle?
4. What is the relationship between polygon sides and their packing efficiency in a circle?
5. How would an octagon compare in terms of packing within the same circle?

**Tip**: When packing shapes into a circle, regular polygons with more sides tend to waste less space.

Explore the geometric process of packing a circle with a 14 cm radius using a square and hexagon. Learn how to calculate the side lengths and areas of the inscribed shapes, applying key formulas and theorems. Suitable for students in Grades 9-11.

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