# How does flying a steeper entry trajectory make the exclusion zone smaller?

This interesting answer Why did SpaceX change the landing site from the Pacific (ITF-1, ITF-2) to the Indian Ocean (ITF-3 and subsequent flights)? contains the following paragraph:

However this type of test creates a situation with three main possible outcomes, a failure to restart the engine, a successful restart but not reaching full duration, or a successful restart with full burn duration. Any of these outcomes will change the location of the landing, so they want to fly a steeper trajectory to keep the exclusion zone as small as possible within all of the various outcomes. This is being achieved by flying the test on a steeper trajectory, with the result being a landing in the Indian Ocean instead of Hawaii.

I'm interested in the highlighted part of the paragraph. How does flying a steeper trajectory make the exclusion zone smaller? I can kind of visualize coming in at an angle and having a elliptical zone of probable impact vs. coming straight in and having a circular zone, but what makes the latter necessarily smaller than the former?

In the sketch, the blue circle is the debris field from the steeper, blue trajectory. We assume the red and blue trajectories are about the same velocity and disintegration occurs at about the same altitude.

The red, high altitude debris cloud at B will be roughly the same area as the Blue debris field at A if distance SA is equal to SB.

The ground level green debris field at C must be larger than the debris cloud at B (and therefor larger than the debris field at A) for 2 geometric reasons:

Even if the field at C was circular, it would be larger in all dimensions than the field at B

The major axis of the almost-ellipse at C is even larger than its minor axis.

However, the danger to humans on the ground depends on the density of people on the ground, not the size of the debris field. The density of falling debris deceases with the square of field diameter, but the exposed population increases by the same proportion. Thus the size of the debris field is less significant than the population density within it. The population density in Africa under ITF-2 was more than 10X the population density in Papua New Guinea, further downrange.

The NASA statement said the effort was to "reduce the size" (cost) "of the exclusion zone", not the risk of human casualties or property damage.