Can the perimeter of a rectangle equal its area?

Answer 1

In one sense the perimeter of a rectangle can never equal its area, because the units are different. If the perimeter is measured in feet, its area is in square feet, and so on.

For that very reason, the equality depends not only on the shape of the rectangle, but also on the choice of units. For a rectangle of any shape, it should be possible to measure it in units that give the desired result.

If you disregard the units, you want solutions for the formula;

xy = 2 (x + y)

Now, assume any particular aspect ratio N, so that y = Nx.

Nxx = 2(x + Nx), or

Nx = 2(1 + N), or

x = 2(1 + N)/N

For the case of a square, N=1 so we have x = 2(2)/1 = 4

Clearly for every nonzero N there is a solution for x.