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Use a ruler to measure the caster's stem. The stem is the metal part that connects the caster to the chair. Also, write down whether or not the stem is threaded or grip.  A threaded caster has a screw that screws into the chair, while a grip stem pulls out like a plug from a socket. If your chair has a swivel plate, measure the width of the plate itself. Using a ruler, measure the wheel itself from the front to the back. Most casters come with 2–2.5 in (5.1–6.4 cm) wheels. If you've had difficulty rolling the chair around (say, onto an elevated surface like a rug), consider buying a larger wheel. A twin-wheel caster tends to last longer, since it distributes weight more evenly, but a single-wheel caster is more reliable, since it has less opportunity to fail, having fewer parts. Twin-wheel casters are more often used in office settings, while single-wheel casters are typically found in industrial settings. Casters can be bought from any hardware store or online hardware store. Be sure to take the specifications you've written down with you, as you'll want to be able to compare them and make sure you're buying the right product. If you're shopping for your casters in-person, take one of the old casters with you and ask someone who works there to help you compare.

Summary:
Write down the size of the caster's stem and what kind of stem it is. Measure the wheel from front to back. Decide whether you want a twin-wheel or single-wheel caster. Purchase your new casters from an office supply store.