How are the distances and diameters of stars measured experimentally? How is the distance of a star and its diameter measured using Earth-based tools? Normally we can measure the angle between the stars edge. But diameter or distance are needed to measure the other parameter. So how do scientists measure both of these parameters?
 A: To measure the size and distance of stars you use a set of tools that build on each other. 
For distance, first there is parallax. Nearby stars have an apparent shift in position relative to distant stars or preferably, galaxies, in the 6 months it takes the Earth to go from one side of the Sun to the other. Triangulation gives the distance.
Then a table of luminance based on star type (spectrum) or behavior (Cepheid and RR Lyre variables which are amazingly consistent from star to star) can be used for stars further away. The variable stars can even be seen in nearby galaxies and their brightness gives a good measure of their distance. 
Knowing the distance from stellar brightness lets you make the next jump to using redshift. You can see the small redshift in local galaxies to calibrate to the current popular value for the Hubble Constant and measurements from there on use redshift. I think some luminance of quasars is used for really great distances, but I am not up to date on that.
For size, stars like the Sun can be used as a reference and some nearby stars were measured with interferometric methods developed (and first performed) by Michelson. Radio astronomy using long baseline interferometry gives very good size info since it depends on coherence length, and therefore size of the source. Elaborations and variations on this have produced size info on stars all over the H-R diagram of stellar mass and luminance.
The work by astrophysics on energy produced in stars and their size versus mass versus spectra has been outstanding and I'm over my head here, but I think those results are of primary use today in determining size. Basically you can give someone the spectra and they can tell you the mass and size of a star.
When we get arrays of optical telescopes in space, direct measurement will be possible on much more distant objects than have been measured so far and this will all get refined by an order of magnitude (or several orders of magnitudes since even planets around stars can be imaged). Note that by "direct measurement" with arrays I really mean optical measurements of something that after a great deal of computation produces an image. 
A: The primary means by which we obtain the distances to stars is through trignometric parallax. The Earth (or a satellite in orbit around the Sun) forms the base of a triangle, with the distant star at the apex, as it orbits the Sun. This changes the apparent position in the sky in a regular, periodic way and this can be modelled to find the "height" of the triangle and hence the distance to the star. All other distance estimation methods are essentially based on calibrations using distances to stars that have been measured in this way.
Because the "parallax angle" gets smaller the more distant the star, then the parallax method gives the most precise results for nearby stars. With the advent of the Gaia astrometry satellite, which has and is measuring extremely precise positions for a billion stars or so, then quite precise parallaxes are now available for of order $10^8$ stars in our Galaxy. These distances can be used to calibrate a host of secondary empirical distance indicators (e.g. how luminous a star really is given its spectral type, a.k.a. spectroscopic parallax; or the relationship between the absolute luminosity and pulsation periods of Cepheid variables or RR Lyrae variables).
Measuring the true radii of stars is harder than measuring parallax and considerably fewer (by many orders of magnitude) stars have direct measurements of their radii. Almost all stars are too far away and too small to be seen as anything but point sources in individual telescopes (though there are  a handful of exceptions in the case of some nearby supergiant stars). This problem is instead finessed using three primary methods.

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*Interferometry. By building an interferometric array of telescopes, separated by a baseline that is much larger than those telescopes, one can attain an angular resolution on the sky that is equivalent to a telescope with a diameter equal to the longest baseline in the array. This is capable of "imaging" or at least detecting how wide a star is in angular terms. The CHARA array is a good example of an interferometer that is often used for the purpose of measuring the angular diameter of stars. If the angular diameter can then be combined with a known distance (from a parallax - see above) then the actual diameter/radius of the star can be calculated. This method works best either for nearby stars or for giant stars, where the angular diameters are large enough to get a precise measurement.


*Eclipsing binary sytems. When two stars orbit their common centre of mass, then if the orbital plane of the system is close to edge-on then we see eclipses in the light coming from the system as one star obscures the other. Using simple gravitational mechanics and geometry then it is possible to determine the ratio of the radii of the stars to their separation. If one can then measure the speeds with which the stars orbit each other, then that can determine their individual masses and separations and leads to a complete determination of the system parameters, potentially with very high precision (see for example Southworth 2020). In principle, this method works even when you don't know the distance to the stars. Unfortunately it can only be applied to the stars in eclipsing binary systems, which are rare, and it is possible that the binary nature of these stars, particularly when the separation is comparable to their radii, means that their radii may not be representative of all stars of their type.


*Occultations. When a luminous, distant star is eclipsed by a "hard edge" in the Solar System (e.g. by the airless limb of the Moon or an asteroid), then the star is not occulted instantly, there is a diffraction pattern formed by the hard edge. The projection of this diffraction pattern moves quickly across the Earth's surface and can be detected with a telescope in the form of a rapid (10-100 Hz) oscillation of the light received from the star as it is occulted. However, because the star has a finite size, then it slides out of view over a finite time and this modifies the diffraction pattern. Analysis of this diffraction pattern leads to an estimate of the stellar angular diameter (e.g. Benbow et al. 2019). The angular diameter can then be converted into a linear diameter if the distance is known via a parallax - see above. To apply this method the star needs to be reasonably bright (or you need a big telescope) in order to get the necessary sampling rate and of course it needs to be occulted by something in the Solar System.
Once direct radius measurements are established for a representative set of stars then one can try to calibrate relationships between radius and luminosity, or radius and surface brightness, perhaps as a function of the temperature and composition of a star and these can be used to estimate the radii of stars that are further away or for which direct radius estimates are unavailable (the majority of stars).
