Depends on where you are and what you are riding.
BMXes usually call it a driver at the back and a sprocket at the front.
Other bikes use different names.
At the rear, if it's only one, it's either a single speed freewheel, or a sprocket.
If it's more than one it's either a cassette or a multi-speed freewheel.
At the front it's called chain wheels or chain rings.
A bicycle designer uses maths to work out the frame angles, and the gearing. The number of teeth on the chainwheels, sprockets (cassette), chain length, and so on, all depend on maths.
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brackets? (square/circle things glued to teeth) wire? (self explanatory)
It's the small toothed wheel that the chain runs over by the rear wheel, although in engineering terms it is any toothed wheel designed for a chain, including the one rotated by the pedal and crank on a bicycle.
circle
what is open kinematic chain?
Usually what happens is that the chain elongates under wear, which forces the rollers farther apart. Depending on set-up this can then cause the chain to ride higher up on the teeth of the sprocket, eventually to the point where it can jump over the top of the teeth. On other set-ups the elongated chain will wear out the flanks of the teeth to the point where their angle can no longer retain the chain.
A chainsaw's chain should be fitted so that the points of the teeth face in the same direction as the chain moves. Then they will cut into the wood without needing much downward pressure. If the teeth are sharp you should never have to force the chain down into the surface of the wood. If you find you have to force the chain down to make it cut the wood then the teeth must either have gone blunt - and the chain needs to be sharpened or replaced - or the chain was installed with the teeth facing the wrong way so that the backs of the teeth just rub over the surface of the wood and you need to press down very hard just to get the teeth to wear down into the wood instead of cutting it cleanly.
Lamprey
The chainring by the pedals and the sprocket(s) by the rear wheel.
Thin sheet metal, yes.
Yes, the pitch circle is an imaginary circle on the gear that passes through the gear teeth and the addendums.