your initial assumption looks good. If the height of the cloud layer above ground, (A) and the elevation angle (a) are known, the distance (c) of the object is at most A/sin(a), the point at which it disappears into the cloud.

Presumably, "A" can be determined from meteorological records and "a" is estimated by the observer. So you can determine the upper limit of the size of the object, but no more.

Determining the distance or size of the object is best done by radian measure, a radian being the angle subtended on a circle by the arc equal in length to the radius of the circle. There are 2 pi radians in a full 360 degrees, so a radian equals 57.3 degrees. A radian arc equals a radius length, so the fraction of a radian subtended by the object equals the fraction of its distance to the observer. Or in general,

Object length = object angle in radians/distance to the object.

So in your example, a ten degree object at 8485' distance is

(10/57.3) x 8485' = 1481' long.

The relationship is linear, so an object of the same angular size at half the distance would be half as big.

Unfortunately, all we can measure is the angular size subtended by the object, not its true distance or true size. A clue as to the distance (like the object is just disappearing into the cloud) can easily give us its distance, hence it's size. But in general, you have to know either the distance or the size in order to get the other. This is why all UFO reports relying on visual estimates of size, speed, distance, etc are not to be trusted.

Subconsciously, an experienced observer can make a very good estimate of distance if he knows the size of the object already, and vice versa. For example, a pilot can say "the 747 was 4 miles away" and he'll probably be pretty close. Other clues can come from nearby landmarks, the tendency of distant objects to show less detail than near ones, and the attenuation of light by haze, etc. But unless the object is a very familiar one to the observer, it's distance, speed and size cannot be determined, and when the witness insists on declaring them, his credibility plummets. Phrases like "it was bigger than any airplane", or "it was only a 100 feet away", or "it was at 10,000 feet and traveling at a thousand miles an hour" are not just wrong, they can't be right.

I can't tell you how many lights I've seen while observing at night which I could not positively identify as a nearby biological (bird or insect), a kite or balloon, aircraft, high altitude meteor, earth-orbital spacecraft, or nearby planetary intruder (close asteroid pass). But there just wasn't enough data to make a decision. If there are real ET spacecraft up there, I've probably already seen one! But I have no way of convincing myself, much less anyone else, that I have.