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There are 22 squares forming a compound shape. Each square has an area of 9cm squared. What is the perimeter of the compound shape

To calculate the perimeter of the compound shape, we will first determine the side length of each square and then analyze the shape's outer boundaries.

1. **Determine the side length of each square:**
   The area of each shaded square is given as \( 9 \, \text{cm}^2 \). The side length \( s \) of a square can be found using the formula for the area of a square:
   \[
   s^2 = 9 \, \text{cm}^2 \implies s = \sqrt{9} \, \text{cm} = 3 \, \text{cm}
   \]

2. **Identify the perimeter of the compound shape:**
   We need to count the number of exposed edges along the perimeter. Since each side of a square is \( 3 \, \text{cm} \), each edge we count will contribute \( 3 \, \text{cm} \) to the perimeter.

Let's analyze the grid:

```
  __ __ __ __ __ __ __ __
 |  |  |  |  |  |  |  |  |
 |__|__|__|__|__|__|__|__|
 |  |  |  |##|##|##|  |  |
 |__|__|__|##|##|##|__|__|
 |  |  |##|##|##|##|##|  |
 |__|__|##|##|##|##|##|__|
 |  |  |  |##|##|##|  |  |
 |__|__|__|##|##|##|__|__|
 |  |  |  |  |  |  |  |  |
 |__|__|__|__|__|__|__|__|
```
Counting the number of exposed edges:

- The top row has 3 exposed squares contributing \( 3 \times 3 \, \text{cm} \).
- The second row has 4 exposed edges (1 on each side of the 3 shaded squares plus the adjacent sides), contributing \( 4 \times 3 \, \text{cm} \).
- The third row has 6 exposed edges (5 sides of the shaded squares plus 1 side of the single shaded square), contributing \( 6 \times 3 \, \text{cm} \).
- The fourth row mirrors the second row, contributing \( 4 \times 3 \, \text{cm} \).
- The bottom row mirrors the first row, contributing \( 3 \times 3 \, \text{cm} \).

Adding these up:
\[
3 + 4 + 6 + 4 + 3 = 20 \text{ exposed edges}
\]

Each edge is \( 3 \, \text{cm} \), so the perimeter is:
\[
20 \times 3 \, \text{cm} = 60 \, \text{cm}
\]

Therefore, the perimeter of the compound shape is \( 60 \, \text{cm} \).

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. What is the area of the compound shape?
2. How would the perimeter change if the squares had different side lengths?
3. Can you calculate the perimeter if the compound shape was made of 25 shaded squares?
4. What if the shaded area was a different polygon, like a rectangle? How would you approach the problem?
5. How does the grid size affect the perimeter calculation?
6. Can you create a formula for the perimeter of such compound shapes?
7. What is the importance of understanding the perimeter in real-life applications?
8. How would you find the perimeter if the shape included partial squares?

**Tip:** When calculating the perimeter of a compound shape, always ensure to count the exposed sides accurately and multiply by the side length.

Learn how to calculate the perimeter of a compound shape composed of 22 squares, each with an area of 9cm². This problem involves determining side lengths, counting exposed edges, and applying basic geometric formulas. Suitable for students in Grades 6-8.

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