To get distances, we use a variety of techniques. The most basic one is geometric parallax. By photographing the same star 6 months apart from points 1 and 2 in earth's orbit, the shift of the star relative to more distant background stars when R = 1 Astronomical Unit amounts to 1 second of arc at 1 parsec ( 3.26 light years), 1/2 arcsecond at 2 parsecs, 1/10 arcsecond at 10 parsecs etc. By the way, at 1 parsec, an arcsecond also subtends 206265 astronomical units.
The Hipparcos astrometric satellite has determined the distance to over 100 thousand stars in this way. Read an ESA Press Release about the mission accomplishments. For example, the distances to the Nearest 10 stars can be found in their Table of 150 closest stars which I reprint below:
Name Parallax Alpha Centauri C 772.33 Alpha2 Centauri C 742.12 Alpha1 Centauri C 742.12 Barnard's Star 549.01 Alpha Canis Majoris (Sirius) 379.12 Epsilon Eridani 310.75 61 Cygni A 287.13 Alpha Canis Minoris 285.93 61 Cygni B 285.42 Epsilon Indi 275.76 Tau Ceti 274.17Note: the Parallax is measured in 1/arcseconds. To calculate the distance in parsecs you have to take 1000.0 and divide it by the parallax number in the last column above. For example, Alpha Centauri C (Proxima) is at a distance of 1000.0/772.33 = 1.295 parsecs which equals 1.295 x 3.26 = 4.22 light years. Alpha Centauri is at 1000/742 = 1.34 parsecs or 4.39 light years. I leave it as a simple calculator exercise for you to convert the parallaxes above into light years!
The result is a detailed map of the stars near the sun out to many parsecs like the figure below.
Stellar diameters can be measured for some nearby giant and supergiant stars by using a technique called stellar interferometry. The Navy Prototype Optical Interferometer has been operating for over a decade at Mount Wilson Observatory, and routinely measures the angular diameters of bright stars to fractions of a milli arcsecond (0.001 arcseconds) accuracy. The table below shows only a few stars that have had their diameters measured. Once their distances are accurately known...from the Hipparcos Survey...their linear diameters in millions of kilometers can easily be found.
The table below shows the sizes in multiples of the solar diameter for some typical stars that have measured angular diameters in column 5 given in arcseconds. The highest resolution of the Hubble Space Telescope is about 0.046 arcseconds. So it is just able to see Betelgeuse as a resolved 'disk'
Name Type dist. diameter Size Alpha Arietis K2III 65.9 ly 0.00699 14.8 Alpha Cassiopeia K0III 150.0 0.00569 27.4 Alpha Persei F5Ib 592 0.00313 59.3 Alpha Leporis F0Ib 1280 0.00177 72.5 Betelgeuse M1Ib 425 0.054 734.4 Antares M1Ib 520 0.041 682.2 Proxima Centauri dM5 4.2 0.007 1.0 Polaris F7 Ib 430 0.00328 45.1The size in kilometers = 3 x 10^13 (d /3.26) (D/3600)/57.3 or 44.6 million x d x D where d = the distance in light years and D is the angular diameter in arcseconds. In terms of solar diameters (1,390,000 km) you get Size = 32 d x D solar diameters. The later formula gives you the above entries in the last column. The super giant star Betelgeuse is 734.4 times the diameter of the Sun.
Here is an image of Betelgeuse showing its actual surface. Note it is not round. Astronomers think this may be due to a large sunspot which dims part of the disk making the shape look irregular.
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