I think part of the calculus is that doing nothing could also trigger a world war, because it would embolden Russia to take even more territory.

Well in general a world War would require of multiple countries on at least 2 divided factions.

If China throws in with Russia, that would be a possible world War.

But with Russia alone, with the sanctions placed on them by the globe, they will likely not be able to sustain a large scale multi front conflict.

"Why is the West confident that helping Ukraine won't trigger WW III?"

They're not. They're fine providing some resources, but are not willing to take other steps. There's no direct involvement, they're not willing to provide stuff like the Polish MIG-29s, etc. The sanctions could provide a cyber war. I'm slightly surprised it hasn't happened yet, but expect that it likely could happen in the next three months.

If it comes down to WW3, then it would end with the carving up of Russia into something more managable like Germany was, plus removing their nukes. Nixon tried the mad man with nukes threat when he was president. Didn't work there either. What we are seeing is Russia's death throes. Communism failed in the late 60's and capitalism saved china when Nixon opened them up in 72. Then Russia went deeper Iron Curtain and plundered and suppressed its populace to 'succeed'. They are suppressing Ukrain because if Ukrain (and Shell) tapped into the Ukrainian oil reserves, they'd be about the #14 oil producing state, and just happened to also be the main pipeline to europe for the USSR. Russia fears that Ukrain could subvert their oil empire. Ukrain will, and already has, because no one will do business with Russia.

I think it's more just that they don't feel they have a choice.

The whole peace in our time thing didn't work out super great last time. At some point you can't just cave indefinitely.

I agree though that I can easily imagine Putin feeling too backed into a corner and bad things happening as a result.

Do you mean global thermonuclear war, or just conventional battle between US/Europe and Russia/China?

What gave you the idea that they are confident?