Seconds of Arc and the Unaided Eye I've been presented with this problem:    


  Say that Jupiter, with its diameter of 142,000 km, was located where Mars now orbits.  What would be the angular size of (the newly-relocated) Jupiter during a close approach, when its distance would be 79,300,000 km?  Would we be able to see Jupiter as a round object with our unaided eye, or only as a point of light?      


The angular size is easy enough to calculate given $ \frac{Angular \space Size}{206,000} = \frac{Linear \space Size}{Distance} $.
So $\frac{Angular \space Size}{206,000} = \frac{142000}{79300000}$ meaning Angular Size = 369 seconds of arc.     

However, would the unaided eye be able to see this? Knowing that the eye can detect 1 minute of arc across, surely  it can detect 6.15? Any help appreciated, thanks!
Yes, you could definitely see Jupiter as a round object if it's 
angular diameter was 369 arcseconds. The limit for seeing something as 
round appears to be a diameter of around 1 arcminute. Notes: 


The most direct comparison is M16, the Eagle Nebula. According to 
http://www.nightskyinfo.com/archive/eagle_nebula/ 


 
The M16 complex is visible to the naked eye on clear nights as a hazy patch 
 

(and thus not a single point), and http://messier.seds.org/m/m016.html 
notes that M16 has an angular diameter of about 7 arcminutes. 


http://www.astropix.com/doubles/ notes that: 


 
In the real world, folks with excellent eyesight can resolve down to 
about 2 arcminutes. Folks with average eyesight can resolve about 3 
arcminutes. 
 

Of course, resolving a double star as two separate points of light 
isn't the same as seeing something as a circle, but provides a good 
upper limit. 


And 
https://en.wikipedia.org/wiki/Phases_of_Venus#Naked_eye_observations 
notes that: 


 
The extreme crescent phase of Venus can be seen without a telescope by 
those with exceptionally acute eyesight, at the limit of human 
perception. The angular resolution of the naked eye is about 1 minute 
of arc. The apparent disk of Venus' extreme crescent measures between 
60.2 and 66 seconds of arc,[4] depending on the distance from 
Earth. Nevertheless it is possible for observers with extremely acute 
eyesight to see a crescent Venus under ideal atmospheric 
circumstances. 
 

Note also that Venus appears circle-like (not point-like) when it transits the Sun, but the same is true of Mercury, so this may be a special situation that doesn't count.