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How high up are the wings of an airplane?
Short answer
On passenger aircraft the wing sits low on the fuselage: on a Boeing 737 the engine nacelles sit about 46 cm off the ground and the wing root just over a meter up. On high-wing light aircraft like the Cessna 172 the wing sits about 2.7 m up. In flight, that wing travels at 9,000-12,000 m of altitude.

Low wing, high wing, and why it matters
The question has two readings and both have concrete answers. The first is the position of the wing relative to the fuselage and the ground. Almost all passenger jets are low-wing: the wing attaches to the lower part of the fuselage, so the wing root sits just over a meter above the runway on a single-aisle aircraft, and the engines hang even lower. On the Boeing 737 that closeness is famous, because the engine nacelles sit about 46 cm (18 in) off the ground; that's why its engines have a flattened intake at the bottom and why the landing gear is deliberately short. The Airbus A320, with somewhat taller gear, leaves more clearance under the engines, and on wide-body aircraft such as the 777 or the A350, the wing starts several meters above the ground and the wingtips exceed the height of a two-story building.
The opposite layout, the high wing, is found on training aircraft and on many regional and military transport planes. On a Cessna 172 the wing sits above the cabin and its upper surface is about 2.7 m up, the aircraft's total height. On an ATR 72 or a C-130 Hercules the wing sits on top of the fuselage so that the engines and propellers stay clear of unpaved runways and so the cargo floor sits almost at ground level.
What determines wing height
Placing the wing low has mechanical advantages: the landing gear anchors into the wing itself and ends up short and light, the fuel and engines stay close to the ground for maintenance, and the wing structure passes below the cabin without taking up passenger space. In exchange, it forces the engines to sit very close to the runway and gives a slight lateral instability that's corrected with dihedral, the upward angle of the wings. The high wing offers natural stability, better downward visibility and clearance for operating on improvised terrain, but it requires landing gear anchored to the fuselage, which ends up longer and heavier, or side fairings to house the wheels.
The other height: how high the wing flies
The second reading is flight altitude, and here the wing travels wherever the aircraft does. Commercial jets cruise between 9,000 and 12,000 m (30,000-40,000 ft), because at that height the air has less than a third of the density it has at sea level and drag drops in the same proportion. The wing still generates the same lift thanks to the higher true airspeed, and the engines burn much less fuel per kilometer. Piston-engine light aircraft, without a pressurized cabin, stay below 3,000-4,000 m, and regional turboprops cruise around 6,000-7,500 m. That's the actual height at which the wings sit for most of the time they're airborne.
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