Using the ratio of the area of a landmass (continent or island) to the elevation of the highest point as a metric, what are the steepest and flattest landmasses on Earth?
In order to compare the two metrics, one must do a bit of thinking about how to relate the two to produce a measure of how steep something is. As an example, imagine a square island with vertices at (0,0,0), (2,0,0), (0,2,0), and (2,2,0), with a peak at (1,1,1). One can easily see that this island has the same general slope as that of a similar island with vertices at (0,0,0), (4,0,0), (0,4,0), and (4,4,0), with a peak at (2,2,2). To get a consistent measure of average steepness, we must compare the square root of the area with the elevation of the highest point. Notice that the units in this make sense too. Using meters for both quantities, we can produce a very reasonable logarithmic graph. Since the difference between logs is the log of the ratio, ln b-ln a=ln(b/a), points along the same slope-1 diagonal line, thus having the same difference, represent landmasses that have the same ratio of highest point elevation to square root of area, or landmasses of the same steepness.
Note that this measure of steepness increases with unreliability as the size of landmasses increase. With more area, one can more frequently take a path that goes up, then down, then up, then down, then up, and this measure calculates steepness as a vector quantity, not as a scalar quantity as would more fit the definition of steepness. At the areas of mountains in continents, steepness is likely drastically larger than reported on the graph, due to the effect of averaging out flat lands in other portions of the continent.
Also note how amazingly flat Arctic lands are.
Here’s a key to the above chart.