Bicycle chains don't actually stretch like a piece of leather. What actually happens is the holes that contain the chain's pins elongate over time (go from a circle to an oval) and the cumulative effect makes the chain longer in length.
To check your chain, you can buy a chain stretch gauge for $7-8 bucks or simply use the tape measure or ruler you already have laying in your tool box.
First, some chain basics (click on any image to enlarge)
The pins on a bicycle chain are spaced 1/2 inch apart. One outer link and one inner link comprise a link-pair with the first and third pins spaced by 1 inch.
Depending on who you consult, a chain should be replaced when it has stretched by more than 0.5%. That means 0.0025 of an inch for a single link. Sure a micrometer can measure that small amount, but there's no good place on the chain to place such a precise instrument!
Instead, we can measure cumulative stretch, that is the total stretch of the chain among many links since all their small deviations will add up to an amount easier to measure. To keep the math simple, we will look for greater than 1/16 inch stretch over 12 link-pairs before replacing the chain.
Ok, get out your tape measure or steel ruler. For a tape measure, we'll start at the 1 inch mark since the hook at the tape's end is too sloppy to line up. Place the 1 inch mark of your ruler precisely at the left edge of a pin near the rear derailer.
From the 1 inch mark, we go another 12 inches to the 13 inch mark. A good chain will have the left edge last pin of the 12th link-pair fall within 1/16 inch of the 13 inch mark.
A worn bicycle chain will show the same pin more than 1/16 inch from the 13 inch mark. Note this chain is almost 1/8 inch too long!
In case you're wondering "What's the big deal if my chain stretches a little? My derailer adjusts for length." Yes, that's true - your rear derailer does adjust for chain length, but NOT for chain link spacing. In other words, the teeth on your derailer, your cassette cogs, and chain-ring teeth have a fixed spacing for, you guessed it, 1/2 inch chain links. Using a worn chain will wear out your other components prematurely.
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