How this instrument works
Given only a rectangle's area and perimeter, with neither individual dimension measured, both length and width can still be recovered — but doing so needs a genuinely different technique than plain division, since two unknowns and only two totals require solving a small system. Setting length + width = P⁄2 (half the perimeter) and length × width = A (the area), the two dimensions turn out to be exactly the two roots of the quadratic equation t² − (P⁄2)t + A = 0.
Solving that quadratic with the quadratic formula gives length = P⁄4 + √((P⁄4)² − A) and width = P⁄4 − √((P⁄4)² − A) — the larger root taken as length, the smaller as width, by convention rather than by any deeper mathematical distinction between the two.
Not every area-and-perimeter combination describes a real rectangle. The term under the square root, (P⁄4)² − A, must be zero or positive; if the perimeter is too small relative to the area, no real rectangle with those two totals can exist, since a fixed perimeter puts a hard upper limit on how much area a rectangle can enclose (that limit reached exactly by a square).
- Enter the rectangle's known area into the Area field.
- Enter the rectangle's known perimeter into the Perimeter field.
- Read Length and Width: the sheet solves the underlying quadratic and reports the larger root as length, the smaller as width.
Worked example — area 48, perimeter 28
A rectangle has an area of 48 and a perimeter of 28. Half the perimeter is 14, so length + width = 14 and length × width = 48 — solving the quadratic t² − 14t + 48 = 0 gives roots 8 and 6, so length = 8 and width = 6 (check: 8+6=14 and 8×6=48, both confirmed).
An area of 25 with perimeter 20 gives length = width = 5 exactly — a square, the special boundary case where the term under the square root reaches exactly zero and the two roots of the quadratic collapse into a single repeated value.
Questions
How do you find a rectangle's dimensions from just its area and perimeter?
Set up length + width = half the perimeter and length × width = the area, then solve for the two dimensions as the roots of the resulting quadratic equation, t² − (P⁄2)t + A = 0, using the quadratic formula.
Why does this need the quadratic formula instead of plain division?
Because there are two unknowns (length and width) and only two known totals (area and perimeter) that combine them nonlinearly — area is a PRODUCT of the two dimensions, not a sum, which is exactly the structure a quadratic equation is built to solve.
What if no real rectangle matches the given area and perimeter?
This happens when the perimeter is too small for the stated area — a fixed perimeter caps how much area a rectangle can enclose, with a square reaching that maximum. If the area exceeds what's geometrically possible for the given perimeter, no real solution exists.
How do I know which result is the length and which is the width?
By convention, this calculator reports the larger of the two roots as the length and the smaller as the width — but mathematically, the two labels are interchangeable, since swapping them doesn't change the rectangle's area or perimeter at all.
What does it mean if length and width come out equal?
The rectangle is actually a square — the special case where the term under the square root reaches exactly zero, meaning both dimensions are forced to the same value, P⁄4, by the given area and perimeter together.