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A teaspoon is an item of cutlery, a measuring instrument, of approximately 5ml, or a unit of measurement of volume (usually abbreviated tsp.).
A barrel is one of several units of volume applied in various contexts; there are dry barrels, fluid barrels (such as the UK beer barrel and US beer barrel), oil barrels and so on. For historical reasons the volumes of some barrel units are roughly double the volumes of others; volumes in common usage range from about 100 to 200 litres (22 to 44 imp gal; 26 to 53 US gal).