Rigel is beta Orion so I am not sure what the question is about.
Rigel and Betelgeuse don't orbit each other. They are hundreds of light-years away from each other.
No. Rigel is in Orion is is pretty far from Polaris.
Betelgeuse is much larger than Rigel. Betelgeuse has a radius about 15 times the radius of Rigel, and a volume of about 3500 times Rigel.
Betelgeuse is about 391.53 light years from Rigel.
Here's a bit of mathematics about working out this distance:
The distance ("a") from Earth to Betelgeuse is about 427 light years and the distance ("b") from Earth to Rigel is about 772 light years.
The angle between the two stars from Earth is 18.56 degrees.
Knowing these three numbers, we can now use the "law of cosines" to calculate the distance (c) between the stars: c = square root of (a2+ b2- 2ab x cosine of the angle).
(In fact, we don't really know the distances to these stars as accurately as this answer may seem to suggest.)
Most of the names of stars are taken from Arabic. Vega came from a tansliteration of "Al Nesr al Waqi" meaning "the falling eagle". Waqi later transliterated to W(V)ega means "falling".
Rigel came from the Arabic name Rijl Jauzah al Yusrā, "the left leg (foot) of Jauzah(Orion)". Rijl laer transliterated to Rigel means "leg, foot".
Betelgeuse is probably older, but it's hard to give exact ages for stars. Both of them are only a few million years old, much younger than our 4 billion year old sun.
See related questions for more details on the ages of those stars.
Rigel is a star and stars do not have a surface, as such. The nearest description of what would be its surface would be a mass of very hot plasma.
Simply because Rigel is hotter.
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It is common for astronomers to identify stars in a constellation with Greek alphabets. This is called the "Bayer designations". Bayer, a German astronomer, designated the stars in a constellation according to brightness, but this is not always the case. Here, Betelgeuse is not as bright as Rigel but it is called Alpha Orionis, rather than Beta Orionis. One reason for this is because 400 years ago, nobody could accurately determine which star was brighter, and it probably seemed that Betelgeuse was the brighter star.
Sirius has a luminosity of: 25.4 while Rigel has a luminosity of: 66,000
Making Rigel brighter than Sirius.
Rigel has exhausted its hydrogen and has swollen out. It is now a blue-white super-giant.
Rigel is the brightest star in the constellation Orion and the sixth brightest star.
It has a radius of 78 times that of our Sun.
You could therefore fit about 474,552 Suns inside Rigel.
See related link for a pictorial difference.
Rigel has a mass about 17 times that of our Sun.
This equates to a mass of 6.726081977 x 1031 kg