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Most tracks these days are 400 meters, just short of a quarter of a mile. Some old or legacy measured tracks remain the Imperial distance of 440 yards (a quarter of a mile) and some tracks are constructed at other distances to fit into available space.
The way to tell if you are on a 400 meter track or something else is to look at how the start lines in other lanes are positioned. Because all levels of the sport now use metric measured tracks, if they are evenly staggered from the common finish line, you are on a 400 meter track (or proportionately half sized 200 meter indoor track). If the staggered start lines are not marked at the same place a race would finish, your distance to the finish line is the difference between the start line in lane one and the common finish. If that distance is a little over 8 feet (2.33m) in front of that common finish line, then you are on an imperial (measured in yards) track.
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ReneNASCAR uses Goodyear Eagle racing slicks for all their major circuits (intermediate and lower still use Hoosier). The width of the current tire is 11.5 inches, however the circumference is altered depending on the track. That is one reason that Goodyear performs so many tests at different tracks throughout the season, to determine the needed tire compound and the final diameter of the tire for each particular track. The most popular size at the track this year ('09) has been the 17 inch diameter. Using the formula (2)(pi)(r), you can determine the overall circumference for the current tire: 2*pi*8.5 (the radius is half the diameter) = 53.4 inches.
EDITED--I believe you made a typo and then ran with it. I think the diameter of the tire is more likely 27". I looked and found sizes from 29 to 32" in diameter for stock cars. If we assume that 27 is the number you should have used, then you get 2*pi*13.5 = 84.82".
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