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:
If you know the width, it's easy to find the length by rearranging this equation to get
If you know the length and want the width, rearrange to get
Example: The area of a rectangle is 20 square meters, and its width is 3 meters. How long is it?
Using the expression
you get
The Square, a Special Case
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.
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,
and the second is that for perimeter,
To solve for one of the variables – say W – you have to eliminate the other.
Since P = 2L + 2W, you can write
You know A = L × W, so
Substituting for W, you get:
Multiply both sides by L to eliminate the fraction, and you get this equation:
This is a quadratic equation, which means it has two solutions derived from the standard formula for solving these equations: The solutions are
Knowing the perimeter may not give you a unique answer, but two answers are better than none.
References
About the Author
Chris Deziel holds a Bachelor's degree in physics and a Master's degree in Humanities, He has taught science, math and English at the university level, both in his native Canada and in Japan. He began writing online in 2010, offering information in scientific, cultural and practical topics. His writing covers science, math and home improvement and design, as well as religion and the oriental healing arts.