
verteilen
to distribute
To hand something out to several people or bring it to different places.
Der Lehrer verteilt die korrigierten Klassenarbeiten an die Schüler.
The teacher distributes the corrected class tests to the students.
to distribute / spread
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to distribute
To hand something out to several people or bring it to different places.
Der Lehrer verteilt die korrigierten Klassenarbeiten an die Schüler.
The teacher distributes the corrected class tests to the students.

to spread
To spread something over a surface so that it covers it more or less evenly.
Bitte verteilen Sie die Sonnencreme gleichmäßig auf der Haut.
Please spread the sunscreen evenly over the skin.

to disperse
To spread out over an area or space; to move apart.
Nach dem Konzert verteilte sich die Menschenmenge schnell in der ganzen Stadt.
After the concert, the crowd quickly dispersed throughout the city.
'Verteilen' (to distribute/spread) is an inseparable verb with the prefix 'ver-'. It is derived from the verb 'teilen' (to divide/share). The prefix 'ver-' here often describes the process of bringing something to different places or people.
Die Hilfsgüter wurden an die Bevölkerung verteilt.
The relief supplies were distributed to the population.
When you state who receives something, use the preposition 'an' with the accusative. The structure is: 'etwas an jemanden verteilen' (to distribute something to someone).
Die Flugblätter wurden an die Passanten verteilt.
The flyers were distributed to the passersby.
To describe the surface on which something is spread out, you use the prepositions 'auf' (on, with dative) or 'über' (over, with accusative).
Verteile die Sauce gleichmäßig auf der Pizza.
Spread the sauce evenly on the pizza.
The reflexive form 'sich verteilen' (to disperse/spread out) is often used for groups of people or things that spread out in a room or area. It describes movement from one point to many.
Nach der Veranstaltung verteilten sich die Besucher in den umliegenden Cafés.
After the event, the visitors dispersed into the nearby cafés.
to share / divide
Teilen emphasizes breaking a whole into several parts or portions. Verteilen, by contrast, emphasizes that the parts reach several people or places. Whoever creates portions teilt them; whoever gives them to the recipients verteilt them.

Mia verteilt die acht Kuchenstücke an die Gäste.
Mia distributes the eight pieces of cake to the guests.
Mia teilt den Kuchen in acht gleich große Stücke.
Mia divides the cake into eight equal pieces.