How do you find the actual diameter of a star going from arcseconds to kilometers?
ex the supergiant star betelgeuse (in the constellation orion) has a measured angular diameter of 0.044 arcsecond. its distance has been measured to be 427 light-years. what is the actual diameter of betelgeuse?

What you have is a skinny, skinny sector of a circle ... like a slice of pie.
The radius of the pie is 427 light-years, and the angle in the center, at
the tip of the slice, is 0.044 second.  That's about  0.000012 degree. 

The question is:  What's the length of the crust out at the fat end of the slice ?

This is a case where it's very handy to measure your angles in radians
instead of degrees. That way, whatever fraction of a radian is at the tip,
the same fraction of the radius is the length of the arc (the crust).

To change degrees to radians, multiply by  (pi)/180 .

0.044 second = 0.000012 degree = about  2.133 x 10^-7 radian .

So the diameter of the star is about  (2.133 x 10^-7) of 427 light years.

I'll leave that part for you to finish up.

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