die Viertelstunde
quarter hour
A period of exactly 15 minutes.
Ich warte schon eine Viertelstunde auf den Bus.
I've been waiting for the bus for a quarter of an hour.

quarter hour
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quarter hour
A period of exactly 15 minutes.
Ich warte schon eine Viertelstunde auf den Bus.
I've been waiting for the bus for a quarter of an hour.

The word is made up of 'Viertel' (quarter; one part of four) and 'Stunde' (hour). So it literally means 'a fourth part of an hour'.
Eine Viertelstunde hat 15 Minuten.
A quarter of an hour has 15 minutes.
'Viertelstunde' (quarter hour) is often combined with time prepositions to describe periods of time: 'in einer Viertelstunde' (in a quarter of an hour; future), 'vor einer Viertelstunde' (a quarter of an hour ago; past), or 'für eine Viertelstunde' (for a quarter of an hour; duration).
Der Film beginnt in einer Viertelstunde.
The film starts in a quarter of an hour.
When giving the time, 'Viertelstunde' is often shortened to 'Viertel' and used without an article. You say 'Viertel nach drei' (quarter past three; 3:15) or 'Viertel vor vier' (quarter to four; 3:45).
Wir treffen uns um Viertel nach acht.
We're meeting at quarter past eight.
The 'akademische Viertelstunde' (academic quarter hour) is a tradition at many universities in German-speaking countries. An event that begins at 10:00 c.t. (cum tempore) actually does not start until 10:15.
Achtung, der Professor beginnt seine Vorlesung immer pünktlich und ohne die akademische Viertelstunde.
Careful, the professor always begins his lecture on time and without the academic quarter hour.
quarter / district
'das Viertel' and 'die Viertelstunde' can both mean a period of 15 minutes. 'Viertel' is shorter and often appears in time expressions, for example 'ein Viertel warten' (to wait a quarter of an hour), or regionally when telling time. 'Viertelstunde' is clearer and means exactly 15 minutes. Learners may confuse the words because 'Viertel' has several meanings: it can also be a district, a quarter of something, or an amount of a drink. When the context is clearly about time, 'Viertelstunde' is often the safer and more precise form.

Das Kind darf noch eine Viertelstunde spielen.
The child may play for another quarter of an hour.
Sie hat ein Viertel gewartet und ist dann nach Hause gegangen.
She waited a quarter of an hour and then went home.