Mountains and waterfalls: height above what?
A mountain height is a distance above sea level, which is itself an average of a moving surface, measured to a summit that may be rock or may be snow. Every part of that sentence has been argued over, some of it for more than a century.
Above what, exactly
Every mountain height in this catalogue, and in almost every atlas, is a height above mean sea level. That phrase hides a surprising amount of work.
The sea is not level. It bulges and dips by tens of metres around the planet in response to gravity, which itself varies with the density of the rock underneath. So mean sea level is not the sea, it is a modelled surface that would coincide with the sea if the sea were undisturbed, and that model has been revised repeatedly. A mountain height is a distance from a mathematical surface to a lump of rock, and both ends have moved over the last century.
Everest, revised in 2020
Mount Everest is quoted at 8,848.86 m, a figure agreed by a joint survey published in 2020 after teams from both countries that share the summit measured it independently. The previous widely used figure was about a metre lower and had stood since the mid twentieth century.
The disagreement that preceded it was not about surveying error. It was about snow. The summit carries a cap of snow and ice of variable thickness, and the question of whether to publish the height of the rock or the height of the snow has been argued for a very long time. The 2020 figure includes the snow, and the rock beneath is a few metres lower.
Mont Blanc's summit is ice, and it is re-surveyed. The figure used here is 4,808 m. The dome of ice on top changes thickness from year to year, so the official height is measured again periodically rather than treated as fixed. Against Everest the ratio is 1.84.
Height is not the same as how big it looks
This is where mountain figures diverge most sharply from experience. Everest rises to 8,848.86 m above sea level, but it stands on a plateau that is already several thousand metres up, so the visible rise from its base is a fraction of the headline number.
Mount Fuji at 3,776 m does the opposite. It is a free-standing cone with no range around it, rising from near sea level in a single sweep. That is why it reads as taller than its number suggests and why it dominates every photograph it appears in.
Fuji against Ben Nevis at 1,345 m gives 2.81. Ben Nevis is the highest point in the British Isles and the walk from the car park covers almost all of it, which makes it one of the few mountains in the set where the stated height and the experienced climb are nearly the same distance.
Waterfalls, where the water does not reach the bottom
Angel Falls is quoted at 979 m, the total drop from lip to the base of the cliff. It is the tallest uninterrupted waterfall there is, and much of the water never arrives as water: it breaks into mist partway down and reaches the ground as rain over a wide area.
That makes the figure a measurement of a cliff rather than of a column of water, which is the only version that can be measured at all. Against the Empire State Building at 443.2 m the ratio is 2.21, comparing a cliff face to a building measured to its antenna tip.
Victoria Falls is the opposite problem. The figure here is 108 m, and that is the highest section, not an average. The falls run more than a kilometre and a half wide and the drop varies along that width by tens of metres, so any single number is a choice about which part to measure. The ratio of 9.06 against Angel Falls is therefore a tallest-section against a total-drop, which is worth stating rather than glossing.
The dimension nobody quotes
Height is a poor summary of a waterfall, because what makes a waterfall impressive is flow, and flow does not appear in any of these figures. Victoria Falls moves an enormous volume over a wide sheet; Angel Falls moves comparatively little over an extraordinary drop. Comparing the two by height alone answers a question almost nobody was asking.
The same limitation applies to mountains in a different form. A height says nothing about mass, and the mass of a mountain is the thing that determines almost everything else about it, including how much it deforms the crust beneath and, through that, where mean sea level sits nearby. The measurement depends on the object being measured, which is the kind of loop that makes geodesy interesting.
Depth, the height nobody sees
Grand Canyon National Park spans 210 km east to west. That figure appears on the comparison with Switzerland at 345.9 km, a ratio of 1.65, and it describes the least interesting property of the place.
The canyon is over a mile deep. No outline drawn flat on a page can carry that, and no span can either. It is the clearest example in the whole catalogue of a measurement being correct, useful for the purpose it was chosen for, and utterly failing to convey what the thing is.
The general rule for natural features
Ask what the zero is. For a mountain it is a modelled sea surface. For a waterfall it is the base of a cliff that may or may not be where the water lands. For a canyon there is no zero at all, only a rim and a floor that both move along its length.
Natural objects do not come with drawings, which is the real difference from a building. A building has a design that fixes where the measurement stops. A mountain has to be argued into a number, and the argument is usually more interesting than the number.