If the circle is the ideal shape for a district, it's because it has an interesting mathematical property: a circle has the shortest possible border containing a given area. Any other shape with the same area will have a longer border. So as far as a metric of physical "compactness," it would seem that you could simply compute the border length/area ratio. Lower numbers are more compact. Actually, to nomalize when comparing districts of different physical sizes you'd want to user border length squared. The most compact possible district (the circle) would then score 4*pi or about 12.6, and less compact districts would score higher numbers. A square distict would be a 16 for example. An equilateral triangle about 20.8 A rectangular district 10 times as tall as it is wide would score a 48.4.

That is, in fact, one common measurement of compactness. As Alan Miller pointed out, however, it can penalize districts that have corrugated edges due to the placement of census tracts. (Reminds me of those fractals with finite area and infinite perimeter.)

FYI most congressional districts are not drawn on census tracts! This is due to strict scrutiny on the equal population requirement. Census blocks is what they are drawn on, and those are not nearly as funky looking as local jurisdictional boundaries, which have to be respected in most states! So I don't know where circles would make sense aside from some obscure math conference.