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wnoise comments on Existential Risk - Less Wrong
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Not much smaller than the earth at all!
With more physics and attention, one could produce better numbers, but as a crude ballpark (using data from wikipedia):
Surface area of the Earth: 510,072,000 km^2
Circumference of ring, if it's placed at 1 AU: 2 * pi AU = 939,951,956 km
So, if the ring is a little over a half a kilometer in width, it has the same surface area as the Earth - and could be smaller still, if we just compare habitable area.
The scale of curvature there makes it clear it's not 1 AU in radius.
Fair enough, I suppose. But then it's not really a ring world so much as a... what? Space station?
Yeah, pretty much. If it were bigger, I might call it a Culture orbital.