Find 1 parsec in millions of km.

Question

Astronomers measure large distances in parsecs. (One parsec is the radius of a circle for which an arc of angle 1/3600° has length equal to the distance from Earth to the sun, about 150 million kilometers.)

Answer 1

Yes, a sketch of a very skinny triangle is useful with tiny angle a (1°/3600 = 1"), the opposite side r being 1 AU = 150 million km, and the adjacent side and hypotenuse both essentially the same distance: let the hypotenuse d = 1 parsec [1 parsec is the distance where the parallax of a star equals 1 second of arc].

It is also expedient to know the small angle approximation, which REALLY applies in this case: sin a ≅ a when any small angle is measured in radians. So convert 1 arcsec into radians: (1° / 3600) * (2π rad / 360°) = 1/206265 radians. Since sin a = a = r/d, d = r/a = (150 million km) / (1 rad/206265) = (150 million km) * (2.063 x 10⁵) = 3.09 x 10⁷ million km.