Look at any long-haul flight on a tracking app and the route appears to bulge toward the pole. London to Los Angeles arcs high over Greenland. Johannesburg to São Paulo dips toward the Antarctic. Passengers assume a detour. Pilots know better: that curve is the straight line.
The chart is lying to you
The confusion comes from projections. The Earth is (nearly) a sphere; your chart is flat. Something has to give. On a Mercator chart, a line of constant direction — a rhumb line — plots as straight, which made life easy for sailors steering by compass. But a rhumb line is not the shortest distance. The shortest distance between two points on a sphere lies along a great circle: the circle whose plane passes through the centre of the Earth.
Stretch a string between two cities on a globe and it settles onto the great circle naturally. On the Mercator chart, that same path plots as a curve bowed toward the nearer pole — the projection distorts it, not the geometry.
The catch: your track keeps changing
Fly a great circle and your true track changes continuously. Follow the string on the globe from A to B and watch the angle it makes with each meridian you cross: it shifts, because the meridians themselves converge toward the poles. That change is called convergency, and for mid-latitudes it's approximated by:
convergency ≈ ΔLongitude × sin(mean latitude)
Two stations 40° of longitude apart at 50°N give a convergency of about 40 × 0.766 ≈ 30.6°. Your initial great-circle track and your final one differ by roughly that amount. This is exactly why great-circle flying only became practical with gyro and inertial reference systems — a magnetic compass wants to fly rhumb lines, and steering a continuously changing heading by hand is miserable.
How much distance does it actually save?
Over short legs, almost nothing — below a few hundred miles, rhumb line and great circle are practically the same track, which is why your VFR nav log doesn't care. Over oceanic distances the difference is real money. London–Tokyo along the rhumb line is roughly 350 nm longer than the great circle. At long-haul fuel burn, that's tonnes of fuel and half an hour of flight time, every single day, on one route.
The exam angles
For ATPL General Navigation, the pattern to internalise: a great circle always lies on the polar side of the rhumb line between the same two points (they coincide only along meridians and the equator). Departure — the east–west distance along a parallel — shrinks with latitude:
departure (nm) = ΔLong (min) × cos(latitude)
And the deeper subtlety from these study notes: the Earth isn't even a sphere. It's an oblate ellipsoid, about 0.3% flatter at the poles, which is why modern navigation systems compute geodesics on the WGS-84 ellipsoid rather than true great circles. For the exam the spherical model is what's tested — but your FMS is doing something slightly cleverer.