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.