By Chris Deziel, Updated Aug 30 2022
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Knowing a rectangle’s area gives you the product of its length (L) and width (W), but it doesn’t uniquely define each dimension. In most cases you need one additional piece of information—such as the other side, a perimeter, or the knowledge that the shape is a square—to solve for both sides.
The basic relationship is A = L × W. Rearranging gives:
L = A ÷ W or W = A ÷ L.
Example: If the area is 20 m² and the width is 3 m, the length is L = 20 ÷ 3 = 6.67 m.
For a square, L = W, so A = L². Thus L = √A.
Example: A square with area 20 m² has side length √20 ≈ 4.47 m.
When the perimeter (P) is also known, you can solve the system:
A = L × W and P = 2L + 2W.
Solving for one variable and substituting into the other leads to the quadratic equation:
2L² – P L + 2A = 0.
Using the quadratic formula gives two possible lengths:
L = [P + √(P² – 8A)] / 2 or L = [P – √(P² – 8A)] / 2.
Once L is found, W can be calculated via W = A ÷ L. These two solutions correspond to the two ways a rectangle can be oriented.
