Sun's angular size = 1920 arcseconds (about half a degree, or the size of your little fingernail held at arm's length).
Most stars are so far away from the Sun that their angular size is not resolved, even in large telescopes. They appear as points of light. Only one star other than the Sun has been seen as a disk, rather than a point. The Hubble Space Telescope made a direct image of the star Betelgeuse. The image has an angular size of only 0.125 arcseconds.
If the angular size `a' of a star and the distance `d' to that star are known, then the radius `R' of the star can be computed.
For trigonometry fans only: the equation is
R = d tan (a/2)
The Sun:
Betelgeuse:
If the star Betelgeuse were superimposed on the Solar System, it would stretch as far as the orbit of Saturn.
A star is approximately a black body. The luminosity of a black body depends very strongly on temperature. Consider a square meter of the surface of a black body. The rate at which it radiates light is given by the following equation:
In the above equation, E is the luminosity of a square meter of the black body's surface, the Greek letter sigma stands for a constant which has been measured in the laboratory, and T is the temperature of the black body's surface.
Double the temperature of a black body, and you increase its luminosity by a factor of 2 x 2 x 2 x 2 = 16.
To find the total luminosity of a black body, multiply the luminosity per square meter by the number of square meters on its surface. A star is well approximated by a SPHERICAL black body (surface area = 4 pi R^2), so the formula which gives the total luminosity of a star is the following:
Thus, if we know the luminosity L of a star (found from its intensity and its distance) and if we know the temperature T of a star, we can compute its radius R.
For example, take the star Sirius (please):
As another example, Sirius has a dim companion, which can be seen through a reasonably large telescope. The companion of Sirius has a luminosity equal to 1/10,000 that of Sirius itself.
The companion of Sirius:
One square meter of the companion of Sirius is (2.6)^4 = 46 times more luminous than one square meter of the Sun. The reason why the companion to Sirius has such a low total luminosity is that it's tiny - even smaller than the Earth.
The largest known star has a radius 3700 times that of the Sun. On the other hand, the objects known as `neutron stars' have radii of only 10 kilometers, although they are even more massive than the Sun.
