Joint work with Akos Dobay, John C. Kern, Kenneth C. Millett, Michael Piatek, Patrick Plunkett, and Andrzej Stasiak. We explore the shape of random polygons by measuring the average dimensions of smallest boxes, spheres, and polyhedra that enclose the polygons. We present computer simulations to examine the differences between these dimensions for polygons with constrained and unconstrained topology. For each measurement, we find that the scaling profiles for polygons of a particular knot type intersect the scaling profile for phantom polygons. The number of edges at which the profiles intersect is known as the equilibrium length with respect to the given knot type and spatial measurement. These equilbrium lengths are then compared to equilibrium lengths with respect to other spatial measurements mentioned here and computed elsewhere.