  We show that a smooth unknotted curve in R^3 satisfies an isoperimetric
inequality that bounds the area of an embedded disk spanning the curve in terms
of two parameters: the length L of the curve and the thickness r (maximal
radius of an embedded tubular neighborhood) of the curve. For fixed length, the
expression giving the upper bound on the area grows exponentially in 1/r^2. In
the direction of lower bounds, we give a sequence of length one curves with r
approaching 0 for which the area of any spanning disk is bounded from below by
a function that grows exponentially with 1/r. In particular, given any constant
A, there is a smooth, unknotted length one curve for which the area of a
smallest embedded spanning disk is greater than A.
