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# Can a perimeter be zero?

Rectangle has following:

x + 3 = w

2x-6 = l

No you can't have a perimeter of 0. The perimeter of a rectangle is given by the formula P = 2L + 2W where L and W are the length and width.

If P = 0 this would imply one of two things

1. L and W are both 0, which in this case you no longer have a rectangle, rather a single point in space.

2. L = -W or W = -L. We know this can't happen because it implies we have a rectangle with a negative length (or Width).

If we consider your rectangle with sides  x + 3 and 2x - 6 and we assume the perimeter is 0 then we get:

P = 2(x + 3) + 2(2x - 6) = 0

2x + 6 + 4x - 12 = 0

6x - 6 = 0

6x = 6

x = 1

Plugging back in for x would result in our sides be 4 and -4 which falls under case 2 which is a no no.

Yes you can have a perimeter of zero. E.g., if the length of the side of a square was defined as a function of time, s(t) = sin(t) + 1, then the perimeter would be zero every time t = 3pi/2 + (2pi)n, n integer. This is relied upon in calculus to ensure the function is continuous.