This question has had the conventional answer given by several people. A
couple of tacit questions have been missed, however, and they need to be
answered for completeness. The first is "How is the mass of Earth
determined?"
Given Newton's spectacularly successful Law of Universal Gravitation,
Henry Cavendish measured the numerical value of "G". He published his
result in a paper aptly titled "Weighing the Earth." That was in 1792.
Having weighed* the Earth, Cavendish had also weighed the Sun, since "G"
appears in Newton's version of Kepler's third law of planetary motion.
Using that law one can also determine the masses of planets with
satellites.
The masses of Venus and Mercury were not known well before they were
visited by space probes (I'm sorry, I don't have the dates at my
fingertips). Since the space probes carried radio transmitters their
accelerations could be determined with great accuracy, and consequently
the planetary masses could also be determined with similar accuracy.
Pluto posed a greater problem. Its mass was estimated by extrapolation
from the brightness observed on Earth, a very uncertain process. This
number had to be revised drastically downward when, in 1978, Pluto was
found to have a satellite, soon afterward named Charon. Soon afterward
the planet and its satellite began a long series of "mutual events",
relative occultations of each body by the other, or part of the other.
As a product of the precision photometry done during those mutual events
the masses of Pluto and Charon were accurately determined, and event
their sizes were measured.
I think that the student who asked this question is on the right track.
We should devote much more time in our teaching to the topics of how we
know what we profess to know, "applied epistemology", one might call it.
Leigh
*"Weighed" means "determined the mass of", if you hadn't guessed.