Understanding the Problem
When the perimeter of a square increases by 25%, we want to determine how this change affects its area.
Step 1: Initial Perimeter and Side Length
- Let the original side length of the square be "s".
- The original perimeter (P) of the square is calculated as:
P = 4s
Step 2: New Perimeter
- An increase of 25% in perimeter means:
New Perimeter = P + 0.25P = 1.25P
- Substituting the original perimeter:
New Perimeter = 1.25(4s) = 5s
Step 3: New Side Length
- To find the new side length (s'), we equate the new perimeter to 4 times the new side length:
5s = 4s'
- Solving for s', we get:
s' = (5/4)s = 1.25s
Step 4: Area Calculation
- The original area (A) of the square is:
A = s²
- The new area (A') with the new side length is:
A' = (s')² = (1.25s)² = 1.5625s²
Step 5: Increase in Area
- The increase in area is:
Increase = A' - A = 1.5625s² - s² = 0.5625s²
Step 6: Percentage Increase in Area
- To find the percentage increase:
Percentage Increase = (Increase / Original Area) × 100 = (0.5625s² / s²) × 100 = 56.25%
Conclusion
Thus, when the perimeter of a square increases by 25%, the area increases by 56.25%.