
der Hof
yard / courtyard
An open area enclosed by buildings or walls, for example the courtyard of a house or the schoolyard of a school.
In der Pause treffen sich alle Schüler auf dem Hof.
During the break, all the students meet in the schoolyard.
yard / farm / court
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yard / courtyard
An open area enclosed by buildings or walls, for example the courtyard of a house or the schoolyard of a school.
In der Pause treffen sich alle Schüler auf dem Hof.
During the break, all the students meet in the schoolyard.

farm
An agricultural business with residential and farm buildings as well as the associated fields and pastures; a farmstead.
Auf dem Hof werden Kühe gehalten und Milch produziert.
Cows are kept and milk is produced on the farm.

court
The residence and entire household of a monarch or prince, including the family, advisers, and servants.
Der französische König Ludwig XIV. war für seinen prunkvollen Hof in Versailles bekannt.
The French king Louis XIV was known for his magnificent court at Versailles.

to court someone
To make an intensive effort to win someone's favor, especially in a romantic context; to woo someone.
Er macht ihr schon seit Monaten den Hof, aber sie zeigt kein Interesse.
He has been courting her for months, but she shows no interest.

court
A short form of Gerichtshof, an institution of the judiciary.
Der Angeklagte wurde vom Hof zu einer Geldstrafe verurteilt.
The defendant was sentenced to a fine by the court.
'Hof' (yard/farm/court) is a base word for many compounds that make the meaning more precise. Examples are 'Bauernhof' (farm), 'Innenhof' (courtyard), 'Schulhof' (schoolyard), or 'Gerichtshof' (court of law).
Die Kinder spielen im Hinterhof des Gebäudes.
The children are playing in the building's backyard.
The preposition depends on the meaning. You are 'im Hof' (in the enclosed courtyard), 'auf dem Hof' (on the farm, in the schoolyard), or 'am Hof' (at the royal court).
Die Hühner laufen frei auf dem Hof herum.
The chickens roam freely around the farm.
The expression 'jemandem den Hof machen' (to court someone) is fixed. It means to woo someone romantically. The person being courted is always in the dative.
Er kauft ihr jeden Tag Blumen und macht ihr den Hof.
He buys her flowers every day and courts her.
When a place is being referred to, key words help distinguish the meaning. 'Schule' (school) or 'Pause' (break) point to a schoolyard, while 'Tiere' (animals), 'Traktor' (tractor), or 'Bauer' (farmer) point to a farm.
Er verbrachte seine Kindheit auf einem Hof auf dem Land.
He spent his childhood on a farm in the countryside.
When 'Hof' (court) describes an institution, words like 'König' (king) or 'Adel' (nobility) point to the royal court. Words like 'Richter' (judge), 'Urteil' (judgment), or 'Angeklagter' (defendant) point to a court of law.
Die Etikette am Hofe des Kaisers war sehr streng.
The etiquette at the emperor's court was very strict.
In the meaning 'königlicher Hof' (royal court), the old dative form 'am Hofe' or 'bei Hofe' is often used. This form sounds historical and is restricted to this context.
Der junge Künstler wurde bei Hofe vorgestellt.
The young artist was introduced at court.
In legal language or in the news, 'der Hof' (the court) is often used as a short form of 'der Gerichtshof' (the court of law). It is a technical abbreviation.
Der Hof befand den Angeklagten für schuldig.
The court found the defendant guilty.
business / operation
For an agricultural business, Betrieb emphasizes the economically organized work. Hof also includes the agricultural site with its residential buildings, farm buildings, fields, and pastures. Betrieb is used for the business, while Hof is used for the entire agricultural property.

Der landwirtschaftliche Hof umfasst ein Wohnhaus, zwei Ställe und mehrere Weiden.
The farm includes a house, two barns, and several pastures.
Der landwirtschaftliche Betrieb beschäftigt zehn Personen und verkauft Milch.
The agricultural business employs ten people and sells milk.