Determining the Length and Width of a Rectangle Using Its Area

If you know the length and width of a rectangle, you can figure out its area. These two quantities are independent, though, so you can’t do a reverse calculation and determine both of them if you know only the area. You can calculate one if you know the other, and you can find both of them in the special case in which they are equal – which makes the shape a square. If you also know the perimeter of the rectangle, you can use that information to find two possible values for length and width.

Determining Length or Width When You Know the Other

The area of a rectangle (​A​) is related to the length (​L​) and width (​W​) of its sides by the following relationship:

(A = L × W)

If you know the width, it’s easy to find the length by rearranging this equation to get

(L = frac{A}{W})

If you know the length and want the width, rearrange to get

(W = frac{A}{L})

​Example: The area of a rectangle is 20 square meters, and its width is 3 meters. How long is it?​Using the expression

(W = frac{A}{L})

you get

(W = frac{20 text{ m}^2}{3 text{ m}} = 6.67 text{ m})

The Square, a Special Case

Because a square has four sides of equal length, the area is given by ​A​ = ​L​2. If you know the area, you can immediately determine the length of each side, because it’s the square root of the area.

​**Example: What are the lengths of the sides of a square with an area of 20 m2?**​The length of each side of the square is the square root of 20, which is 4.47 meters.

Finding Length and Width When You Know Area and Perimeter

If you happen to know the distance around the rectangle, which is its perimeter, you can solve a pair of equations for L and W. The first equation is that for area,

(A = L × W)

and the second is that for perimeter,

(P = 2L + 2W)

To solve for one of the variables – say ​W​ – you have to eliminate the other.

1. Use One Equation to Express One Variable in Terms of the Other

Since ​P​ = 2​L​ + 2​W​, you can write

(W = frac{P – 2L}{2})

2. Substitute This Value in the Other Equation

You know ​A​ = ​L​ × ​W​, so

(W = frac{A}{L})

Substituting for ​W​, you get:

(frac{P – 2L}{2} = frac{A}{L})

3. Rearrange Terms

Multiply both sides by ​L​ to eliminate the fraction, and you get this equation:

(2L^2 – PL + 2A = 0)

This is a quadratic equation, which means it has two solutions derived from the standard formula for solving these equations: The solutions are

(L = frac{P + sqrt{P^2 – 8A}}{2} text{ and } L = frac{P – sqrt{P^2 – 8A}}{2})

Knowing the perimeter may not give you a unique answer, but two answers are better than none.