Carrying engagement ring in carry-on backpack. Must I declare it at the airport customs in Europe? [duplicate]

I think you are above the limit. Look at

https://ec.europa.eu/taxation_customs/individuals/travelling/entering-eu_en

The limit is 430 euros.

Unfortunately, without more information, a lot of the answers to your questions are "it depends". Let me state that I am not a customs agent, but researching this question has raised a few interesting differences in customs systems around the world.

Starting in North America, whether you ought to declare the ring depends on where you are. For example, the US customs form asks for the value of "you are bringing into the United States and will remain in the United States", which obviously doesn't apply in this case. On the other hand, the Canadian instructions for visitors require "you must declare all goods when you arrive at the first CBSA port of entry."

Once you reach the EU, some things get more straightforward. Your allowance in €430, which it seems you exceed. This means that you should declare the ring when it enters the EU (since you're carrying it as hand luggage at the first airport you arrive at), and pay the duty and tax of the relevant country. The actual process will vary depending on which airport you make your connection, as will the tax and duty to be paid(compare the UK and Germany). Once you've done this, you'll have nothing more to declare at your final destination (If you'd put it in your checked baggage instead, then it would be declared here).