By how much would the perimeter of a square be decreased if its area is halved?

To solve this problem, let's first analyze the relationship between the area and the perimeter of a square.

### Step 1: Determine the side length of the original square.
Let the side length of the original square be \( s \). The area of the square is given by:

\[
\text{Area} = s^2
\]

### Step 2: Determine the new side length after halving the area.
If the area is halved, the new area is \( \frac{s^2}{2} \). Let the new side length be \( s' \). Then:

\[
(s')^2 = \frac{s^2}{2}
\]

Taking the square root of both sides:

\[
s' = \frac{s}{\sqrt{2}}
\]

### Step 3: Calculate the perimeters.
- The perimeter of the original square is \( 4s \).
- The perimeter of the new square is \( 4s' = 4 \times \frac{s}{\sqrt{2}} = \frac{4s}{\sqrt{2}} = 2\sqrt{2}s \).

### Step 4: Determine the decrease in perimeter.
The decrease in the perimeter is the difference between the original and the new perimeters:

\[
\text{Decrease in perimeter} = 4s - 2\sqrt{2}s = s(4 - 2\sqrt{2})
\]

### Step 5: Simplify the expression.
The numerical value of \( 4 - 2\sqrt{2} \) is approximately:

\[
4 - 2\sqrt{2} \approx 4 - 2 \times 1.414 \approx 4 - 2.828 \approx 1.172
\]

So, the perimeter decreases by approximately \( 1.172 \times s \).

### Conclusion:
The perimeter of the square would be decreased by approximately \( s(4 - 2\sqrt{2}) \), or roughly \( 1.172 \times s \), where \( s \) is the original side length of the square.

Would you like more details or have any questions?

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1. How does the area of a square relate to its perimeter?
2. What is the exact value of \( \sqrt{2} \)?
3. How does changing the side length affect the perimeter of a square?
4. What is the relationship between area and side length in any polygon?
5. How would the answer change if the area was tripled instead of halved?

**Tip:** When dealing with square roots, it's often helpful to approximate their values to understand the magnitude of the result.

Explore how the perimeter of a square changes when its area is halved. Understand the relationship between area and perimeter in squares with detailed calculations and examples. Suitable for Grades 7-9.

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