How do we know/calculate the exact age of sun ☉ ? ie. 4.57 billion years. What is the way to calculate it?
The Sun was formed about 4.57 billion years ago from the collapse of part of a giant molecular cloud that consisted mostly of hydrogen and helium and which probably gave birth to many other stars. This age is estimated using computer models of stellar evolution and through nucleocosmochronology. The result is consistent with the radiometric date of the oldest Solar System material, at 4.567 billion years ago. Studies of ancient meteorites reveal traces of stable daughter nuclei of short-lived isotopes, such as iron-60, that form only in exploding, short-lived stars. This indicates that one or more supernovae must have occurred near the location where the Sun formed. A shock wave from a nearby supernova would have triggered the formation of the Sun by compressing the gases within the molecular cloud, and causing certain regions to collapse under their own gravity. As one fragment of the cloud collapsed it also began to rotate due to conservation of angular momentum and heat up with the increasing pressure. Much of the mass became concentrated in the center, while the rest flattened out into a disk which would become the planets and other solar system bodies. Gravity and pressure within the core of the cloud generated a lot of heat as it accreted more gas from the surrounding disk, eventually triggering nuclear fusion. Thus, the Sun was born.
I'm not familiar with nucleocosmochronology (mentioned by Gugg), but traditionally the age of the Solar System, and therefore of the Sun, has been estimated by dating meteorites using uranium decay.
The material that makes up the Solar System was formed in supernovae, and this forms the two isotopes of uranium, U-238 and U-235, in an allegedly well known ratio. The two isotopes decay at different rates and as their end products produce different isotopes of lead. So if you take a meteorite and measure the relative amounts of U-238, U-235, Pb-206 and Pb-207, you can estimate the age of the meteorite. The oldest meteorites are aound 4.57Gyrs old so we assume the Sun is this age as well.
i used the word "allegedly" above because I believe there is some debate about the correct U-238 to U-235 ratio, though the argument is over only very minor corrections to the ratio. Also the material that formed the Solar System may have come from many supernovae rather than just one, so there would be a spread of the formation dates.
I don't know a great deal about this area, but the above should be a good basis for Googling for more detail.
We do know that the solar system has formed along with the sun (being a third generation star). There are some isotopes distributed (though rarely) throughout the solar system. In Earth, we can do a radio dating for such decaying elements, which can be found in rocks, meteorite locations, etc..
For confirmation, we can check our observations along other areas of our planet, or some other planet in our own solar system. Now, there are many other methods to determine the age, especially using the Standard model of astrophysicists.
Another way is to have a look at the luminosity and spectra of other stars (star clusters can provide more data, as there is a lump) using the standard parallax measurement, plot them in the H-R diagram and finally compare the observed main-sequence stars with sun, and thereby have a guess on its total age...