Math worksheet for finding the apothem of polygons, including examples and practice problems.

I'll solve this problem by first understanding what an apothem is and then calculating it for each polygon shown in the image.

Looking at the example provided, I can see that the apothem is calculated using the formula:
Apothem = (2 * Area) / Perimeter

This makes sense because for a regular polygon, the area can be calculated as:
Area = (1/2) * Perimeter * Apothem
Rearranging this gives us: Apothem = (2 * Area) / Perimeter

Now I'll calculate the apothem for each of the 9 polygons shown in the worksheet:

1. For the first polygon (hexagon):
- Area = 123.26 ft²
- Perimeter = 48 ft
- Apothem = (2 * 123.26) / 48 = 246.52 / 48 = 5.14 ft

2. For the second polygon (pentagon):
- Area = 250.43 ft²
- Perimeter = 50 ft
- Apothem = (2 * 250.43) / 50 = 500.86 / 50 = 10.02 ft

3. For the third polygon (pentagon):
- Area = 37.25 yd²
- Perimeter = 20 yd
- Apothem = (2 * 37.25) / 20 = 74.5 / 20 = 3.73 yd

4. For the fourth polygon (square):
- Area = 36 yd²
- Perimeter = 24 yd
- Apothem = (2 * 36) / 24 = 72 / 24 = 3.00 yd

5. For the fifth polygon (triangle):
- Area = 125.34 ft²
- Perimeter = 60 ft
- Apothem = (2 * 125.34) / 60 = 250.68 / 60 = 4.18 ft

6. For the sixth polygon (pentagon):
- Area = 207.13 m²
- Perimeter = 50 m
- Apothem = (2 * 207.13) / 50 = 414.26 / 50 = 8.29 m

7. For the seventh polygon (octagon):
- Area = 90.04 ft²
- Perimeter = 32 ft
- Apothem = (2 * 90.04) / 32 = 180.08 / 32 = 5.63 ft

8. For the eighth polygon (hexagon):
- Area = 186.20 ft²
- Perimeter = 48 ft
- Apothem = (2 * 186.20) / 48 = 372.4 / 48 = 7.76 ft

9. For the ninth polygon (hexagon):
- Area = 168.26 yd²
- Perimeter = 48 yd
- Apothem = (2 * 168.26) / 48 = 336.52 / 48 = 7.01 yd

The solution to this problem is to apply the formula Apothem = (2 * Area) / Perimeter for each polygon, which gives us the apothem values as calculated above.