Circles are among the most fundamental shapes in both the natural world and human engineering. Stars, which are spheres (or objects approximating spheres, to be picky), have the ability to give life to planets like Earth. The projection, or geometric shadow, of a sphere is a circle, and both of these forms have innumerable implications in astronomy, mathematics, architecture and elsewhere.
The Unit Circle
A circle can be divided into 360 degrees, or 360°. That is, one "trip" around the circle subtends an angle of 360°; alternatively, 1/360th of the circle is "captured" by a single angular degree.
Each degree, like each hour on a clock, can be divided by 60 to yield minutes (in this case, arcminutes) and then again by 60 to yield seconds. Thus the number of arcseconds in a circle is considerable:
Radians vs. Degrees
Yet another way to measure angles is in radians. This unit of measure takes into account the fact that circles and π are hopelessly intertwined. Because 2π times the radius equals the circumference, circle angles can be measured in radians, with 2π of these making up one full revolution.
Because one full revolution is also 360°, there are 2π radians per 360°, which works out to 360/(2 × 3.14159) = 57.3 degrees per radian. Similarly, 2π radians/360° = 0.017453 radians per degree. To convert from radians to arcseconds, multiply by 206,265 arcseconds per radian.
Whether you choose to work in degrees, radians or arcseconds depends entirely on the parameters and scale of the problem you are given to work through.
Degrees, Minutes and Seconds of Arc
If you are looking at a diagram of a circle on a typical phone screen or even a laptop computer, it would be hard to imagine visualizing what one sliver of that circle would look like if it were divided into 360 pieces, much less 21,600 pieces (the total individual minutes) or well over a million pieces (all of the seconds).
But if you are standing on, say, the Earth, which is about 25,000 miles around, the story changes. Now, 25,000 miles/1,296,000 arcsec = 0.0193 miles per arcsec. Multiplying this by 60 gives 1.16 miles per arcmin, and multiplying again by 60 gives about 69.4 miles per degree. In fact, this is very close to the number of miles in a minute of latitude on the Earth grid coordinate system.
Because lines of longitude converge (draw closer together) between the equator and their meeting at the poles, these lines are not a fixed distance apart, unlike lines of latitude (also called "parallels" for this reason).
The Arcsecond: Earthly and Heavenly Applications
When you look at the sun or the moon, you may think they take up a fair chunk of the sky, maybe a couple of degrees of arc. Instead, each is a disk that happens to take up about 1/2° (1,800 arcsec) of the sky. This figure seems surprisingly low to many people, perhaps because these are the largest objects in the sky despite their objectively modest proportions. It is counterintuitive to imagine 360 suns or moons fitting neatly together to take up the 180° of sky between the horizons, but it would be possible.
This and the above section illustrate the utility of the arcsecond or arcsec: Very small fragments of circles can have considerable proportions if the size of the circle as a whole is sufficiently great!