Dimensions of a Rectangle Calculator

Find the length and width of a rectangle given its area and perimeter by solving a system of equations with the quadratic formula.

Enter Area & Perimeter

Result

Rectangle Dimensions
10 x 5
units
Length (longer side)10
Width (shorter side)5
Area50
Perimeter30
Diagonal11.180340

Step-by-Step Solution

l + w = P/2 = 15, l x w = A = 50

Finding Rectangle Dimensions from Area and Perimeter

A rectangle is defined by two dimensions: length (l) and width (w). If you know the area (A = l x w) and the perimeter (P = 2l + 2w), you can determine both dimensions by solving a system of two equations simultaneously. This is a classic algebra problem that reduces to solving a quadratic equation.

The Mathematical Method

System of Equations

From the area and perimeter, we get two relationships between length and width.

l + w = P/2 and l x w = A

Quadratic Equation

Substituting w = (P/2) - l into l x w = A yields a quadratic in l.

l^2 - (P/2)l + A = 0

Quadratic Formula

Solve using the quadratic formula to find both dimensions directly.

l = (P/2 +/- sqrt((P/2)^2 - 4A)) / 2

Step-by-Step Derivation

Starting from the two equations:

  1. Let S = P / 2 (the semi-perimeter). Then l + w = S.
  2. We also know l x w = A.
  3. Express w = S - l and substitute: l(S - l) = A.
  4. Expand: lS - l2 = A, or equivalently l2 - Sl + A = 0.
  5. Apply the quadratic formula: l = (S +/- sqrt(S2 - 4A)) / 2.
  6. The two solutions give you l and w (the larger value is the length, the smaller is the width).

When Do Solutions Exist?

Real dimensions exist only when the discriminant S2 - 4A is non-negative, where S = P/2. This means (P/2)2 must be at least 4A. Geometrically, this corresponds to the constraint that among all rectangles with a given perimeter, the square has the maximum area. So for a given perimeter P, the maximum possible area is (P/4)2.

  • If S2 - 4A > 0: Two distinct positive dimensions exist (a proper rectangle).
  • If S2 - 4A = 0: The rectangle is actually a square (l = w = S/2).
  • If S2 - 4A < 0: No real rectangle exists with the given area and perimeter.

Practical Applications

This calculation appears frequently in real-world scenarios:

  • Fencing a garden: You have a fixed amount of fencing (perimeter) and want a specific area.
  • Construction: Determining room dimensions from known floor area and wall length.
  • Packaging: Finding box face dimensions given surface constraints.
  • Land surveying: Computing lot dimensions from area and boundary measurements.

Related Rectangle Formulas

  • Area: A = l x w
  • Perimeter: P = 2(l + w)
  • Diagonal: d = sqrt(l2 + w2)
  • Aspect ratio: r = l / w