The mass of one atom of carbon is 12 u (unified mass)
1 u = 1.66 * 10pow(-24)g

hence mass of 1 atom of carbon is 12 * 1.66 * 10 pow(-24)
=> 19.92* 10pow(-24)g

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There are multiple ways.

1. Look into the periodic table. It mentions the relative atomic mass. For carbon (12), it will be mentioned as 12.  Then multiply with the value of atomic mass unit (amu) 1.665 *10^-24 gms.  Then you get the mass of an atom of carbon (12) isotope.

2. Find the number of protons and number of neutrons in the nucleus. For carbon (12) isotope, the numbers are  6 and 6 respectively.
    total = 12   multiply with the mass of protons and mass of neutrons and add.

3.  Take some sample of carbon.  Find its weight.  Using reactions like burning in oxygen, burn all of carbon forming carbon dioxide.  Using spectroscopy, determine the composition of the gas. Find the proportion of CO and CO2 using spectroscopy.  Find the total number of molecules in the gas, using  gas laws.  like PV = nRT.
           The total number of molecules in the gas gives number of atoms of Carbon in the sample.  Divide the total mass of sample by number of atoms.

4. Average atomic mass --  calculation 

Take a sample.  Using spectroscopy, determine the proportions of C(12), C(13), C(14) , C(15) isotopes present in the given sample.  Then multiply the ratios with 12, 13, 14, 15 respectively and add them.
  Multiply this total with the value of  amu = 1.665*10^-24 gms.