How do you find the length and width of a rectangle from area and perimeter?

If you know the area (A) and perimeter (P) of a rectangle, first find the half-perimeter: s = P / 2. This equals length + width. The two dimensions are then the two roots of the quadratic equation x^2 - s*x + A = 0, solved using x = (s +/- sqrt(s^2 - 4A)) / 2. The larger root is the length and the smaller root is the width. For example, with area 20 and perimeter 18, s = 9, and solving the quadratic gives dimensions of 5 and 4.

Can you find both dimensions from area and one side alone?

Yes. If you know the area (A) and one side (say the length, L), the other side (width) is simply W = A / L. This is the direct rearrangement of the area formula A = L x W. No quadratic equation is needed in this case because you already have one of the two unknowns.

What if the area and perimeter given do not produce a real rectangle?

A valid rectangle only exists if the discriminant of the quadratic (s^2 - 4A, where s is half the perimeter) is zero or positive. If the discriminant is negative, no real rectangle has that combination of area and perimeter, because the perimeter is too small to enclose that much area. If the discriminant is exactly zero, the rectangle is actually a square, since both roots are equal.

Dimensions of a Rectangle Calculator

Work out the length and width of a rectangle from the information you already have. Enter the area and perimeter, the area and one side, or the perimeter and one side, and this calculator solves for both dimensions using the standard algebraic method.

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Method verified Standard rectangle area and perimeter formulas: A = L x W, P = 2(L + W).

1. What You Know

Known information

Area (square units)

Perimeter (units)

Known side (units)

2. Units

Unit label (display only, does not affect maths)

Enter area in square units matching your chosen unit (for example square metres if using metres) and perimeter or side length in the same linear unit.

Rectangle Dimensions

Length
-
Longer side

Width
-
Shorter side

Area (check)

Length x Width

Perimeter (check)

2 x (Length + Width)

Method Used

Mode-

Half perimeter (s = P/2)-

Discriminant (s^2 - 4A)-

Equation solved-

Result-

Rectangle Properties

Length-

Width-

Diagonal (sqrt(L^2 + W^2))-

Aspect ratio (L:W)-

Summary: Enter your known values above.

How to Find the Dimensions of a Rectangle

A rectangle has two pairs of equal sides: a length (the longer side) and a width (the shorter side). If you already know both dimensions, working out the area and perimeter is straightforward. This calculator solves the reverse problem: working backwards from the area, perimeter, or one known side to find both dimensions.

Method 1: Area and Perimeter Known

This is the classic version of the problem and it uses a quadratic equation. If the area is A and the perimeter is P:

Find the half-perimeter: s = P / 2. Since P = 2(L + W), this means s = L + W.
You now know that L + W = s and L x W = A. These are the sum and product of the two dimensions.
Both dimensions are roots of the quadratic equation x^2 - s*x + A = 0.
Solve using the quadratic formula: x = (s +/- sqrt(s^2 - 4A)) / 2.
The larger root is the length, and the smaller root is the width.

For example, if the area is 20 and the perimeter is 18: s = 9, and the discriminant is 9^2 - 4(20) = 81 - 80 = 1. So x = (9 +/- 1) / 2, giving x = 5 or x = 4. The rectangle is 5 by 4.

Method 2: Area and One Side Known

If you already know one side (say the length, L) and the area (A), the other side is found directly by rearranging the area formula:

Width = Area / Length

No quadratic equation is required because you already have one of the two unknowns.

Method 3: Perimeter and One Side Known

If you know the perimeter (P) and one side (say the length, L), the other side comes from rearranging the perimeter formula P = 2(L + W):

Width = (Perimeter / 2) - Length

When There Is No Valid Rectangle

When solving from area and perimeter, the discriminant (s^2 - 4A) must be zero or positive for a real rectangle to exist. If it is negative, the perimeter given is too small to enclose that much area, and no real rectangle satisfies both values. If the discriminant is exactly zero, the two roots are equal, meaning the rectangle is actually a square.

Related Calculators

Maths and Stats Calculators: the full hub of maths tools.
Area of a Rectangle Calculator: find area from length and width.
Perimeter Calculator: work out perimeter for common shapes.
Diagonal of a Rectangle Calculator: find the diagonal from length and width.
Rectangle Calculator: full set of rectangle properties.

Method: Standard Euclidean geometry formulas for rectangles: Area = Length x Width, Perimeter = 2 x (Length + Width). The area-and-perimeter case is solved using the quadratic formula applied to the sum and product of the two dimensions.

This calculator assumes a true rectangle (opposite sides equal, all angles 90 degrees). Results are given to two decimal places. Always check that your inputs use consistent units (for example, square metres for area with metres for perimeter or side length).

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