As students progress through their math education, they will eventually learn about geometry, specifically the measurements of polygons. One important concept is the perimeter, which is the distance around the outside of a shape. When a circle is inscribed, or drawn inside a polygon, it can provide an interesting challenge for students to calculate the perimeter. Here are some activities to help teach students this concept.

**1. Create a nested shape challenge**

For this activity, create a series of polygons that have a circle inscribed in them. Have students use string or a ruler to measure the perimeter of each shape. Then, challenge them to create a larger polygon that includes all of the smaller ones, with the same inscribed circle. This helps students understand that the perimeter of a larger shape with the same inscribed circle will be greater than that of a smaller shape.

**2. Draw an inscribed circle**

Give students a polygon and a compass. Have them draw an inscribed circle in the center of the polygon. Then, have them find the length of the sides of the polygon and calculate the perimeter. This helps students understand that the perimeter of a polygon with an inscribed circle is equal to the sum of the sides of the polygon.

**3. Use visual aids**

Provide students with plastic circles and polygons of different sizes. Have them match the circle to the polygon that it would fit inside of. Then, have them measure the perimeter of the polygon and the diameter of the circle. This activity helps students visualize the relationship between the perimeter and diameter of a circle inscribed in a polygon.

**4. Compare inscribed and circumcircles**

For this activity, provide students with a polygon and have them draw an inscribed circle and a circumcircle, which is a circle that passes through all of the vertices of the polygon. Have them measure the diameter of each circle and calculate the perimeter of the polygon with both circles. This activity helps students understand the difference in perimeter when a circle is inscribed versus when a circle is circumcised.

**5. Practice using formulas**

Finally, provide students with a set of polygons and circles, and have them use the formula for finding the perimeter of a polygon (P = n x s) and the formula for finding the circumference of a circle (C = 2πr) to calculate the perimeters of polygons with inscribed circles.

By using these activities, students can gain a better understanding of the perimeter of polygons with an inscribed circle. These hands-on activities help students visualize and apply the concepts they learn, making geometry more engaging and accessible