How was the parsec derived?

"Parsec" is a portmanteau of the words "Parallax" and "arcsecond"; an object one parsec distant has a par allax of one arc sec ond

The definition for a parsec being ~3.26 light-years is essentially trigonometric in origin. If, observed from Earth, an object appears to move a distance of one arcsecond in the sky when is at two opposing points in its orbit (say, June and December, six months apart), then it must be a certain distance away.

In the diagram above, for a parallax angle p of one arcsecond (so ##4.85*10^-6 rad##), we can apply some simple trigonometry to derive the value of one parsec.

The angle ##p=4.85*10^-6 rad##, and the length of the shortest edge of the right triangle formed in the diagram (Sun-Earth line) is ##1AU~=1.5*10^11m##.

So the length of the edge connecting to the nearby star, ##x##, is given by rearranging ##tan(4.86*10^-6 rad)=(1.5*10^11m)/x##

Rearranged for ##x##: ##x=(1.5*10^11m)/(tan(4.85*10^-6 rad))=3.093*10^16m##

This distance happens to be 3.26 light years or so, and as such a new unit was defined to make measuring these distances easier; the parsec.