This is something that has not sat right with me forever that I wish to finally get straight. This is not a homework assignment. Rather, it is something my nagging curiosity needs to resolve: While I understand the concept of similar triangles and the side-splitting theorem, something has never sat right with me when trying to understand the apparent size of distant objects. Looking at this diagram, if the eye is assumed to be at the vertex (marked “a”), then the diagram suggests relative vertical size of an object should increase as the horizontal distance increases. Yet, we know by observation size decreases as distance increases. So, should one ultimately take the reciprocal of the result from the side-splitting theorem? (i.e., double the distance would yield half the apparent size of a distant object.)

Yet, when applying this same theorem to enlarging an image, for example, the side-splitting theorem yields the correct answer without any adjustment since a direct proportion is the proper model in this situation. In short, are they both proper applications of the side-splitting theorem; however, the first case requires an extra step, I presume? Perhaps this one little caveat was missing way back in my 10th grade geometry class?

I’m not following this reasoning. It’s certainly true that if an object moved away from the viewer but maintained the same angular size in his vision, then the object would be increasing in actual size. Is that what you mean?

The side splitter theorem doesn’t seem particularly relevant to this example. The most relevant way I can see to apply it would be to say that various different objects at different distances which all subtend the same angular size to a viewer must have their actual sizes in proportion to the distance from the viewer.

Use the calculator at you link. Click on Angle since that’s what you want to solve for then enter several different distances with the same size. If the calculator works correctly, (and it seemed to for me) the longer the distance, the smaller the angle will be.

Is that not what you observe when you see things at different distances?

Your brain knows that true size is proportional to angular width times distance. Since we generally know how big something is and our eyes give us angular size, we can “solve” intuitively for distance. Or if we know roughly how far away something is, we can solve for actual size.

IOW, we’re used to watching a receding object of fixed size become angularly smaller. Or an approaching object of fixed size become angularly larger.

That intuition works fine. Until you look at the OP’s diagram, which holds angular size constant and lets physical size grow as distance does.

IOW, that’d be equivalent to watching a receding balloon as it was being inflated at just the right rate to subtend the same angle as it recedes and grows.

Which:
A) is something you’ve never seen in your life.
B) would fool your intuition and be confusing to watch.

The OP isn’t watching something physically move and grow or not, but the thought process of reasoning about the diagram is exactly analogous. and that’s where he gets confused.

Make a different diagram which shows an object of fixed size at two different distances and hence subtending two different angles. Same exact underlying reality relating the three variables. We’re just holding a different one constant and watching how the other two react to changes in each other.

No. For a given object, the “Size” stays the same. (That’s the actual, physical size of the object.) If you keep the “Size” the same, and increase “Distance” (i.e. move the object further to the right), you end up with a longer but thinner triangle. The “Angular Size” decreases.

This diagram shows the different heights of objects that make the same apparent ANGLE. Its a graph of height vs distance, for one particular apparent angle.

What you need is the graph of angle vs distance when the HEIGHT of the object remains constant.

As the horizontal distance increases, then the vertical size of the object required to maintain the same visual diameter does increase.

If you put a 10ft tall object 50 yards away, then if you want to put an object 100 yards away and have it appear the same height, then that object will have to be twice as tall, ie 20ft tall.