Being a mile above the ground increases your travel distance by about 1/4000, or 0.025%. This is astoundingly small.
Being a mile above the ground reduces air density by something like 20% so you’re pushing through 20% less air and moving through 0.025% more from extra distance. Extra altitude is clearly the winner.
Thinner atmosphere means less air is in the way of the plane (less air resistance).
It also means less climbs/descents over the course of the flight. Each one is more turbulence, and each climb costs more fuel.
It also means less weather. Sure, there’s *really* tall clouds out there, but much of the flight is above the level of most clouds. This also means less turbulence and better fuel efficiency.
As others have said, the air is thinner and results in less aerodynamic drag. Drag increases approximately with the square (in most, but not all cases- drag is a highly complex calculation that can vary based on many environmental and situational variables) of velocity so, as you can imagine, if you reduce the density of the medium by ~20% you will massively decrease the drag when the object is traveling through that thinner air at hundreds of mph.
The most efficient flight path is climbing at best climb rate up to the midpoint, then descending from there into the destination. In practice, you usually run out of altitude before that happens, and you start cruising near your top altitude instead. Short flights will look like that triangle, though. Take off, climb out, hit ceiling, five minutes later top of descent.
More fuel efficient, especially if a flight can catch a “jet stream” tail wind (available from 30,000 to 40,000+ feet) , more time for pilots to react to failures and emergencies, greater distance to glide to a suitable landing spot gives more landing options in an emergency or engine failure landing, higher altitudes give greater vertical distance and altitude options for the many jets flying in the sky at the same time. (Depending on your heading a flight may be told an odd or even number of thousands of feet to fly at )
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