When do I apply Significant figures in physics calculations?

I'm a little confused as to when to use significant figures for my physics class. For example, I'm asked to find the average speed of a race car that travels around a circular track with a radius of $500~\mathrm{m}$ in $50~\mathrm{s}$.

Would I need to apply the rules of significant figures to this step of the problem? $$C = 2\pi (1000~\mathrm{m}) = 6283.19$$

Or do I just need to apply significant figures at this step? $$\text{Average speed} = \frac{6283.19~\mathrm{m}}{50~\mathrm{s}} = 125.664~\mathrm{m}/\mathrm{s}$$

Should I round $125.664~\mathrm{m}/\mathrm{s}$ to $130~\mathrm{m}/\mathrm{s}$ since the number with the least amount of significant figures is two?

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mathematical-physics probably isn't the right tag for this. But I don't know what is, mathematics isn't appropriate either. – tpg2114 Feb 2 '13 at 0:23
@tpg2114 I agree, I couldn't find an appropriate tag for this question. – Scott Feb 2 '13 at 0:25
Related: physics.stackexchange.com/q/32078/2451 and links therein. – Qmechanic Jun 18 '13 at 18:06

You should always find an answer that is a formula, and then only apply significant figures once you get to the one final step of substituting your numbers back into the problem in place of variables.

Avoid multiple intermediate steps of substituting numbers at all costs. Not only will this save your pencil a lot of work, but it will also cause your answer to be more accurate, as rounding errors can pile up, even when using a calculator.

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So the correct answer would just be 130 m/s then, right? – Scott Feb 2 '13 at 0:23
@Scotty Yes, if the 500 and 50 given to you each had at least 2 sig figs. Some conventions are "50" has one sig fig (so your answer should round to 100) and "50." (note the decimal) is more precise. You should check with your instructor/textbook for what the conventions are. – Chris White Feb 2 '13 at 0:37
This is good advice, but sometimes it makes sense to, e.g., write down an intermediate result so that you can check whether it's reasonable, because it's of some interest on its own, or because you're doing calculations in your head or on paper without your calculator handy. (Or for those of us old enough to remember slide rules ...) In this situation, you need to keep enough extra sig figs to avoid accumulating significant rounding errors. If there are 10 arithmetic operations, then we expect that roughly 1 extra sig fig is needed. If 100 operations, 2 extra sig figs, etc. – Ben Crowell Jun 19 '13 at 0:50
The other thing to point out to the OP is the reason for sig fig rules. The reason is to avoid miscommunicating to other people the precision of your result, e.g., in the CIA fact book that gives the population of Nigera to 8 sig figs. If intermediate steps aren't being communicated to anyone else, then their sig figs aren't an issue. – Ben Crowell Jun 19 '13 at 0:52

Keep precision all the way through to the number you report and then truncate accordingly at the end.

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No, this is silly. What if your intermediate result is $\sqrt{2}$? Are you going to write down infinitely many sig figs, or keep infinitely many sig figs in the memory of your calculator? If you're going to record intermediate numerical results, then it makes sense to keep enough extra sig figs (typically 1 or 2) to keep from accumulating significant rounding errors. – Ben Crowell Jun 19 '13 at 0:46