der Bau
construction
The process of erecting a building or another structure.
Der Bau der neuen Brücke wird voraussichtlich zwei Jahre dauern.
The construction of the new bridge is expected to take two years.

construction / building
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construction
The process of erecting a building or another structure.
Der Bau der neuen Brücke wird voraussichtlich zwei Jahre dauern.
The construction of the new bridge is expected to take two years.

The noun 'der Bau' (construction/building) is directly derived from the verb 'bauen' (to build). This connection helps you understand the basic meaning of erecting something and the structure that results from it.
Wir bauen ein Haus.
We are building a house.
The many meanings of 'Bau' can often be recognized by the preposition. 'Auf dem Bau' means a construction site, 'im Bau sein' means 'to be under construction', and 'in den Bau gehen' is colloquial for 'to go to prison'.
Mein Bruder arbeitet auf dem Bau.
My brother works on a construction site.
The phrase 'im Bau sein' (to be under construction) is often used to express that something is currently being built or created. It can refer to buildings, but also to projects such as a website.
Das neue Einkaufszentrum ist seit einem Jahr im Bau.
The new shopping center has been under construction for a year.
Using 'der Bau' for 'die Baustelle' (construction site) is colloquial. In formal texts or announcements, you are more likely to find the word 'Baustelle'.
Auf dem Bau wird heute nicht gearbeitet.
No work is being done on the construction site today.
In the context of animals, 'der Bau' refers to their underground dwelling. It is often used in compound words such as 'Fuchsbau' (fox den) or 'Dachsbau' (badger sett).
Der Dachs verschwand in seinem Bau im Wald.
The badger disappeared into its den in the forest.
The term 'der Bau' for 'das Gefängnis' (prison) is a casual, colloquial expression. It is often used in films or informal conversations.
Er saß zwei Jahre im Bau.
He spent two years in jail.
building
'Gebäude' and 'Bau' can both mean a structure built by people. 'Gebäude' is the normal, neutral word for a house or a larger building with walls and a roof. 'Bau' can also mean a building, especially if it is large or striking. Very often, however, 'der Bau' means the process of construction, that is, that something is currently being built. This is why learners may confuse the words. If you are simply talking about a finished house, office, or museum, 'Gebäude' is usually clearer.

Der Bau des neuen Krankenhauses beginnt im nächsten Frühjahr.
Construction of the new hospital begins next spring.
Am Bahnhof zeigt sie ihrer Familie das alte Gebäude aus rotem Stein.
At the train station, she shows her family the old building made of red stone.