7. Describe a physical model that you could use to illustrate the concept of a finite area being enclosed by an unlimited perimeter. Copyright © 2015 Pearson Education, Inc. All rights reserved.

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To illustrate the concept of a finite area being enclosed by an unlimited perimeter, you can use the physical model of a **circle** with an infinitely thin boundary. Here’s how you can describe this model step by step:

1. **Materials Needed**:
   - A large piece of paper or a whiteboard.
   - A compass or a round object (like a cup) to trace a circle.
   - A ruler to measure the radius.

2. **Creating the Circle**:
   - Use the compass to draw a circle with a specific radius $$ r $$. Alternatively, trace around a round object to create a circle.
   - Label the center of the circle as point $$ O $$.

3. **Understanding the Area**:
   - The area $$ A $$ of the circle can be calculated using the formula:
     $$
     A = \pi r^2
     $$
   - This area is finite and represents the space enclosed within the circle.

4. **Understanding the Perimeter**:
   - The perimeter of the circle, also known as the circumference $$ C $$, can be calculated using the formula:
     $$
     C = 2 \pi r
     $$
   - As the radius $$ r $$ increases, the circumference increases linearly, but it remains finite for any given radius.

5. **Illustrating the Concept of Unlimited Perimeter**:
   - To illustrate the concept of an unlimited perimeter, you can discuss the idea of a fractal boundary, such as the **Koch snowflake**. 
   - Start with an equilateral triangle and iteratively add smaller triangles to each side. As you continue this process infinitely, the perimeter increases without bound, while the area enclosed remains finite.

6. **Conclusion**:
   - This model effectively demonstrates that while the area can be finite (like the area of a circle or the area of the Koch snowflake), the perimeter can become infinitely large through certain constructions or as the radius increases.

By using this physical model, you can visually and mathematically convey the concept of a finite area being enclosed by an unlimited perimeter.
You can use a circle to show a finite area with an unlimited perimeter. Draw a circle with a specific size. The area inside is limited, but if you make the circle bigger, the edge (perimeter) keeps getting longer without end.

Describe a physical model that you could use to illustrate the concept of a finite area
being enclosed by an unlimited perimeter.
Copyright © 2015 Pearson Education, Inc. All rights reserved.

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