How this calculator works
Area fixes the product of the sides and half the perimeter fixes their sum. The two sides are therefore the roots of a quadratic. A negative discriminant means no real rectangle fits. The longer root is found first, then the area divided by that root gives the shorter side, reducing cancellation for thin rectangles.
Formula
l + w = P/2; lw = A; t² - (P/2)t + A = 0; maximum possible A = P²/16. Area efficiency = A/(P²/16).
Worked example
Area 12 m² and perimeter 14 m give a side sum of 7 m. The roots of t² - 7t + 12 are 4 and 3. A square with that perimeter would have area 12.25 m².
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Frequently asked questions
Can area and perimeter give two different rectangles?
Only swapping the side names changes the answer. The unordered side pair is fixed.
Why is the maximum area a square?
For a fixed side sum, the product is largest when the two sides are equal.
What does efficiency mean here?
It compares geometric area with the square of the same perimeter. It does not measure material waste or building performance.