circumstance
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circumstances
Usually in the plural: the conditions that accompany and influence a particular situation or event.
Unter den gegebenen Umständen ist das die bestmögliche Lösung.
Under the given circumstances, this is the best possible solution.
The word is made up of 'um' (around) and 'Stand' (standing/position). Literally, it describes what “stands” around a situation — that is, the accompanying conditions.
Der wichtigste Umstand war, dass wir nur wenig Zeit hatten.
The most important circumstance was that we had very little time.
The word 'Umstand' is almost always used in the plural ('die Umstände'), whether it means 'Bedingungen' (conditions) or 'Mühe' (trouble/effort). The singular is rare and rather formal.
Die Umstände haben sich geändert.
The circumstances have changed.
In the meaning of 'Bedingungen' (conditions), the word is very often used with the preposition 'unter' (+ dative).
Unter normalen Umständen würde ich das nicht tun.
Under normal circumstances, I wouldn't do that.
The meaning 'Mühe' (trouble/effort) occurs almost only in the fixed expression '(sich) keine Umstände machen' (not to go to any trouble). It is a polite phrase.
Das Essen war schnell gemacht, ich habe mir keine großen Umstände gemacht.
The meal was quick to make; I didn't go to much trouble.
The expression 'in anderen Umständen sein' is a euphemistic and somewhat old-fashioned paraphrase for 'schwanger sein' (to be pregnant). You are more likely to find it in older texts or when someone wants to be very discreet.
Sie erzählte uns, dass sie in anderen Umständen sei.
She told us that she was expecting.
case / fall
'der Fall' and 'der Umstand' can both have to do with a situation. 'der Fall' refers more to a specific situation or a concrete event: you look at something as an individual example or problem. 'der Umstand' refers more to the conditions surrounding a situation, that is, what influences it. That is why people often say 'in diesem Fall' (in this case), but 'unter schwierigen Umständen' (under difficult circumstances). Learners confuse the words because both point to situations. A practical tip: 'Fall' is the situation itself; 'Umstände' are the surrounding conditions.
Unter diesen Umständen können wir den Termin leider nicht halten.
Under these circumstances, we unfortunately cannot keep the appointment.
In diesem Fall müssen wir zuerst die Chefin informieren.
In this case, we have to inform the boss first.