What are the rectangles that have the same perimeter as area?

Perimeter is a length, and a length cannot be the same as an area.

Ignoring the units, all rectangles that have the same numerical perimeter as area are those that satisfy:

2 x (length + width) = length x width

which can be rearranged to give:

width = 2 x length/(length - 2)

Meaning that given any length over 2, a width can be found to give a rectangle that meets the requirement that its numerical perimeter is the same as its numerical area. (For a length greater than 0 and less than 2, the width would be negative and not possible; similarly for a length less than 0, the length is negative and not possible. When the length is 2, the width is undefined and so not possible. When the length is 0 the width is 0 and it is not a rectangle.)

At some stage as the length increases, the length will equal the width and as the length continues to increase the rectangle then given will match the previous rectangles with the length and widths swapped. This occurs when:

length = 2 x length/(length - 2)

⇒ length x (length - 4) = 0

⇒ length = 4.

So, as long as the length is greater than or equal 4 it will be the longer side - the length, by convention, is the longer side. Thus all rectangles satisfy:

width = 2 x length/(length - 2)

with length ≥ 4, will have the numerical value of their perimeter the same as the numerical value of their area.

For example:

• 4 cm x 4 cm: perimeter = 16 cm, area = 16 cm2
• 5 cm x 31/3 cm: perimeter = 162/3 cm, area = 162/3 cm2
• 6 cm x 3 cm: perimeter = 18 cm, area = 18 cm2
• 41/2 cm x 33/5 cm: perimeter = 161/5 cm, area = 161/5 cm2
• etc

Note: the first example is a square which is a rectangle with all the sides the same length.