February Question:
A seamstress is working with a square pillow pattern. She begins to wonder what might happen to the area of her pillow pattern if she increases the perimeter of the original pattern.
How will increasing the perimeter of a square or rectangle affect the area of the polygon? Consider:
- If you double the length and width of the square/rectangle, how will the new area compare to the original area?
- What if you triple or quadruple the length and width?
- Can you find a pattern for how any increase in length and width will affect the area of the original and new figures?
Answer:
Only one formula is needed to solve this problem: Area = length x width
For this problem, it may be helpful to examine a figure with labeled dimensions:
| 2 cm |
 |
4 cm | Area = length x width Area = 4 cm x 2 cm Area = 8 cm 2 |
If you double the length and width, your new figure looks like:
| 4 cm |
 |
8 cm | Area = length x width Area = 8 cm x 4 cm Area = 32 cm² |
When we compare the area of the two figures, the first has an area of 8 cm² and the second an area of 32 cm². If we tried this with rectangles of other dimensions, we would see the same results. When we double the perimeter, the area of the new figure will be 4 times larger than the original figure.
What if we triple the perimeter?
| 2 cm |
 |
4 cm | Area = length x width Area = 4 cm x 2 cm Area = 8 cm² |
The new figure would have a length of 12 cm and a width of 6 cm.
| Area = length x width Area = 12cm x 6 cm Area = 72 cm² |
When we compare the new areas, we find that the figure with the tripled perimeter has an area that is 9 times greater than the original area.
What if we quadruple the perimeter? The new figure would have a length of 16 cm and a width of 8 cm.
| Area = length x width Area = 16 cm x 8 cm Area = 128 cm² |
When we compare the new areas, we find that the figure with the quadrupled perimeter has an area that is 16 times greater than the original area.
Let’s put the data in a chart:
Perimeter Increase |
Area |
X 2 |
X 4 |
X 3 |
X 9 |
X 4 |
X 16 |
It would be safe to assume that if we increase the perimeter of a rectangle 5 times, the area would be 25 times greater.
The pattern is the perimeter increase, n, squared, n².