Which has a smaller perimeter with the same area of 36 square units, a regular Pentagon, a regular hexagon, a regular octagon

To compare the perimeters of a regular pentagon, hexagon, and octagon that all have the same area of 36 square units, we first need to express the area and perimeter formulas for these polygons.

### Area and Perimeter Formulas

1. **Regular Pentagon:**
   - **Area (A):** \[ A = \frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} \times s^2 \]
   - **Perimeter (P):** \[ P = 5s \]

2. **Regular Hexagon:**
   - **Area (A):** \[ A = \frac{3\sqrt{3}}{2} \times s^2 \]
   - **Perimeter (P):** \[ P = 6s \]

3. **Regular Octagon:**
   - **Area (A):** \[ A = 2(1 + \sqrt{2}) \times s^2 \]
   - **Perimeter (P):** \[ P = 8s \]

Where \( s \) is the side length of the polygon.

### Steps to Compare Perimeters

1. **Solve for the side length \( s \) in each case using the area:**
   \[
   s_{\text{pentagon}} = \sqrt{\frac{4A}{\sqrt{5(5 + 2\sqrt{5})}}}
   \]
   \[
   s_{\text{hexagon}} = \sqrt{\frac{2A}{3\sqrt{3}}}
   \]
   \[
   s_{\text{octagon}} = \sqrt{\frac{A}{2(1 + \sqrt{2})}}
   \]

2. **Plug these side lengths into the perimeter formulas:**
   \[
   P_{\text{pentagon}} = 5 \times s_{\text{pentagon}}
   \]
   \[
   P_{\text{hexagon}} = 6 \times s_{\text{hexagon}}
   \]
   \[
   P_{\text{octagon}} = 8 \times s_{\text{octagon}}
   \]

Let's calculate these step by step.The perimeters for each polygon with an area of 36 square units are as follows:

- **Regular Pentagon:** Approximately 22.87 units
- **Regular Hexagon:** Approximately 22.33 units
- **Regular Octagon:** Approximately 21.84 units

### Conclusion
The **regular octagon** has the smallest perimeter when compared to the regular pentagon and hexagon for the same area of 36 square units.

Would you like more details or have any questions?

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Here are some related questions you might find interesting:

1. How do the side lengths compare for these polygons with the same area?
2. What is the relationship between the number of sides of a polygon and its perimeter for a given area?
3. How would the perimeters compare for polygons with different areas?
4. Can this comparison be extended to a regular dodecagon (12 sides)?
5. How does the shape of the polygon affect its efficiency in enclosing area?
6. What would happen if the area was increased to 100 square units?
7. How does the ratio of the perimeter to the area change for these polygons?
8. What are some real-world applications where minimizing the perimeter for a given area is important?

**Tip:** Regular polygons with more sides tend to have smaller perimeters for the same area, approaching the characteristics of a circle.

Explore how the perimeters of regular pentagons, hexagons, and octagons with an area of 36 square units compare. Includes detailed calculations and conclusions. Suitable for advanced high school students.

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