Anxiety has an inconvenient property: it is faster than thought. By the time it appears, the body is already responding - heart quicker, breath shallow and high in the chest, muscles tight - and only afterwards comes the explanation of why. Which is why reasoning that there is nothing to panic about does not really work. You are arguing with a part of yourself that does not take arguments.
What is reachable is the body. And the fastest route into it is the breath.
Why breathing works at all
Breathing is the only function of the autonomic nervous system that runs by itself and can also be changed deliberately. You cannot order your heart to slow or your blood pressure to drop; you can order a longer exhale, and the heart follows.
The mechanics are simple. An inhale slightly speeds the pulse, an exhale slows it. A breath where the exhale is longer than the inhale therefore tells the body something thought cannot: that there is no danger. The response is not instant and not complete, but it is reliable, and you can verify it on yourself.

Three exercises
One: four in, eight out. Inhale through the nose for a count of four. Exhale through the nose for a count of eight. If eight is too much, start at six. Ten repetitions, anywhere, including sitting at a desk.
This is the base exercise and the only one worth practising daily, including when you do not need it. The reason: in a moment of anxiety you will not remember anything new. Only what is already a habit works.
Two: breathing with a hand on the belly. One palm on the chest, the other on the belly. Breathe so the lower one moves and the upper one stays still. Do not count, just observe. Fifteen breaths.
This is for moments when counting itself adds tension - for some people counting is one more thing that has to be done correctly. A hand on the belly is more direct and easier to hold on to.
Three: a long exhale through pursed lips. Inhale through the nose, exhale through slightly pursed lips, as if blowing at a candle without putting it out. Let the exhale run as long as it will, without forcing the end. Eight repetitions.
This is the most useful one in an acute moment, because it gives a clear physical sensation - the resistance at the lips is something to focus on when everything else is too fast.
Which one, when
Do the first one preventively: five minutes in the morning or evening, every day. Not because it helps in that moment, but so that when you do need it, the body already knows how it goes.
Do the second when the anxiety is quiet and sustained - that tightness in the chest that lasts all day without a trigger.
Do the third when it is acute, escalating, and you need something that works within a minute.

What to expect
Not calm in the sense of the anxiety disappearing. What realistically happens is that the peak is lower and shorter. Instead of forty minutes in which the state only escalates, it is ten or fifteen before it starts to turn.
The first few attempts often fail, and that is normal. Breathing under anxiety is hard, because breath is precisely what is most disrupted. That is why the exercise is learned while you are calm.
In Qi Gong this is trained systematically: an hour of slow breathing under mild physical load. It is why people who train regularly find the long exhale available even when things are difficult - not through discipline, but because the body has already done it three hundred times.
Where the limit is
Breathing exercises are not treatment for an anxiety disorder. If anxiety limits your work, relationships or sleep, if you have panic attacks, or if you avoid things you used to do, that is territory where professional treatment helps - and it is effective.
Breath is a tool worth having then too. But it is not a substitute, and we do not present it as one.

