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What are passenger aircraft called?
Short answer
Passenger aircraft are called commercial aircraft or airliners. They are classified by size into regional (up to about 100 seats), narrow-body or single-aisle (100-240 seats, such as the Airbus A320 or Boeing 737) and wide-body or twin-aisle (250-500 seats, such as the A350, 787 or 747).

Airliners, commercial aircraft or jetliners
The technical term is passenger transport aircraft, though everyday usage favours commercial aircraft or airliner. Aviation certification groups them under the transport category (FAA Part 25 and EASA CS-25), which sets structural and safety requirements for aircraft over 5,700 kg. Each model also carries its manufacturer's name and a number: Boeing uses the 7X7 series (707, 737, 747, 787), Airbus the A3XX series (A320, A350, A380) and Embraer the E-Jet designation (E175, E195). Variants are identified with suffixes, such as 737-800 or A321neo, where neo stands for new engine option.
The three families by size
Regional aircraft seat between 50 and 100 passengers and cover short routes: turboprops such as the ATR 72 or the De Havilland Dash 8, and jets such as the Embraer E175 or the Airbus A220. Narrow-body or single-aisle aircraft, with 120 to 240 seats, are the most numerous in the world; the Boeing 737 and the Airbus A320 family together total over 20,000 units delivered and operate most domestic and European flights. Wide-body aircraft have two aisles and 250 to 500 seats, and fly intercontinental routes: Boeing 777 and 787, Airbus A330 and A350, and the now-discontinued 747 and A380, the only ones with two full decks.
Other names you'll hear
Jumbo was originally applied to the 747 and by extension to any wide-body aircraft. Jet designates an aircraft powered by jet turbines, as opposed to a turboprop. Charter is not a type of aircraft but a flight booked outside regular scheduling. And business jets, such as Gulfstreams or Falcons, carry passengers but are not considered airliners.
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