How to find the length of a side of a rectangle with variables

Updated November 03, 2020

By Chris Deziel

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.

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

A = L \times 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}

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

Because a square has four sides of equal length, the area is given by A = L2. 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.

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 \times 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.

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

W = \frac{P - 2L}{2}

You know A = L × W, so

W = \frac{A}{L}

Substituting for W, you get:

\frac{P - 2L}{2} = \frac{A}{L}

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.

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The rectangle below has a perimeter of 54. Find the lengths of the unknown side. That is, find S.

Possible Answers:

Explanation:

For a rectangle, its perimeter is the sum of all for sides. We can write

Simplify

Solve for S

Given a rectangle with a length

units longer than its width and an area of

square units, find the length of the rectangle's shortest side.

Possible Answers:

Correct answer:

Explanation:

The given rectangle has a length that is

units longer than its width. This can be expressed in the following equation, where

is the length and

is the width of the rectangle.

Since the area of the rectangle is equal to its length multiplied by its width (

), and the area of the rectangle is given, the following equation must be true.

Replacing

in this equation with its value stated in the first equation results in the following.

Distribute the variable into the parentheses.

Factor the polynomial.

and

are both solutions for this equation, but

is not valid as a width for a rectangle. The width of the rectangle is

units, which is the shorter side since the length is

units longer

There is a regtangular fence surrounding a park. The perimeter of the fence is

. What is the width of the fence if the length is

Possible Answers:

Correct answer:

Explanation:

The formula for the perimeter of a rectangle is:

, where

represents the length and

represents the width.

We know the perimeter of the rectange is

and the length is

. Plugging these values into the equation, we get:

A rectangle has a perimiter of 36 inches and a length of 12 inches. What is the width of the rectangle in inches?

Possible Answers:

Correct answer:

Explanation:

To find the width, multiply the length that you have been given by 2, and subtract the result from the perimeter. You now have the total length for the remaining 2 sides. This number divided by 2 is the width.

A rectangle four and a half times as long as it is wide has perimeter of 272 centimeters. To the nearest tenth of a centimeter, how wide is it?

Possible Answers:

Correct answer:

Explanation:

If we call the width of this rectangle

, then its length

can be restated as

, or, equivalently,

The perimeter can then be written as:

Since the perimeter of the rectangle is 272 cm, we can set up the following equation:

The width, in cm, of a rectangular fence is 2 more than half its length, in cm. Which of the following gives the width, w cm, in terms of length, l cm, of the rectangular fence?

Possible Answers:

Correct answer:

w = ½ l + 2

Explanation:

To find the width, we must take half of the length, which means we must divide the length by 2. Then we must take 2 more than that number, which means we must add 2 to the number. Combining these, we get:

w = ½ l + 2

The width of a rectangle is 2 inches longer than 3 times its length. Which of the following equations gives the width, w, of the rectangle in terms of its length, l,?

Possible Answers:

Correct answer:

w = 3l + 2

Explanation:

The width equals 3 times the length, so 3l, plus an additional two inches, so + 2, = 3l + 2

Your dad shows you a rectangular scale drawing of your house. The drawing is 6 inches by 8 inches. You're trying to figure out the actual length of the shorter side of the house. If you know the actual length of the longer side is 64 feet, what is the actual length of the shorter side of the house (in feet)?

Possible Answers:

Explanation:

We can solve this by setting up a proportion and solving for x,the length of the shorter side of the house. If the drawing is scale and is 6 : 8, then the actual house is x : 64. Then we can cross multiply so that 384 = 8x. We then divide by 8 to get x = 48.

If the perimeter of a rectangle is

and the length of the rectangle is

, what is the width of the rectangle?

Possible Answers:

Correct answer:

Explanation:

Recall how to find the perimeter of a rectangle.

We can then manipulate this equation to find the width.

Now, plug in the information given by the question to find the width.

If the perimeter of a rectangle is

and the length of the rectangle is

, what is the width of the rectangle?

Possible Answers:

Correct answer:

Explanation:

Recall how to find the perimeter of a rectangle.

We can then manipulate this equation to find the width.

Now, plug in the information given by the question to find the width.

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