
das Viertel
district
A part of a city; a neighborhood or a specific district.
Wir wohnen in einem sehr lebendigen Viertel mit vielen Cafés.
We live in a very lively district with lots of cafés.
quarter / district
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district
A part of a city; a neighborhood or a specific district.
Wir wohnen in einem sehr lebendigen Viertel mit vielen Cafés.
We live in a very lively district with lots of cafés.

quarter
A part of a whole that corresponds to one of four equal parts (¼).
Ich hätte gern ein Viertel von dem Apfelkuchen.
I'd like a quarter of the apple cake.

quarter hour
A quarter of an hour (15 minutes), often used when telling time.
Es ist jetzt Viertel nach vier.
It's now quarter past four.

quarter liter
A quantity for drinks equal to 0.25 liters, especially for wine.
Zum Essen bestellte er sich ein Viertel Rotwein.
With his meal, he ordered a quarter liter of red wine.
The word 'Viertel' (quarter) is derived directly from the number 'vier' (four). All meanings relate to the idea of a 'fourth part'—whether a part of the city, a quarter of a whole, a quarter of an hour, or a quarter liter.
Für das Rezept brauche ich nur ein Viertel Liter Milch.
For the recipe, I only need a quarter liter of milk.
The context makes the meaning clear: In connection with a city, it refers to a district. With 'nach' (past) or 'vor' (to/before) and a time, it means a quarter of an hour. When ordering wine, it means a quarter liter.
Wir treffen uns um Viertel vor acht.
We're meeting at quarter to eight.
When referring to a district, 'Viertel' (district) is often used with prepositions such as 'in' (in) or 'durch' (through). Typical combinations are 'in einem Viertel wohnen' (to live in a district) or 'durch ein Viertel spazieren' (to stroll through a district).
Ich wohne in einem ruhigen Viertel am Stadtrand.
I live in a quiet district on the outskirts of the city.
For times, 'Viertel' (quarter) is used in fixed expressions: 'Viertel nach' (quarter past) for 15 minutes after the full hour and 'Viertel vor' (quarter to) for 15 minutes before the full hour.
Der Zug fährt um Viertel nach fünf ab.
The train departs at quarter past five.
In parts of eastern Germany and Austria, you often hear a different time expression: 'dreiviertel' (three quarters) followed by the next hour means 'Viertel vor' (quarter to). For example, 'dreiviertel acht' means the same as 'Viertel vor acht' (7:45).
Mein Termin ist um dreiviertel zehn.
My appointment is at quarter to ten.
district
Viertel and Bezirk can both refer to part of a city. Viertel is more often an urban residential area or locality. Bezirk is usually an officially defined administrative area, so it is especially appropriate in administrative contexts and when fixed boundaries are involved.

Das Viertel rund um den Markt ist bei Familien beliebt.
The neighborhood around the market is popular with families.
Der Bezirk, zu dem der Markt gehört, hat eine eigene Verwaltung.
The district that includes the market has its own administration.