r = 1AU gives a surface of 12.57 square AU (that unit boggles my mind as I write this)
r = 3.5 AU for Betelgeuse and the surface area is 153.94 AU. That 3.5 puts the sphere at the surface of the star.
The 8-20 cm thick surface becomes roughly 0.8 - 2 cm.
Betelgeuse is roughly 100,000 times more luminous than the sun (it is a variable star so I'm using a very round number).
The habitable zone is a function of luminosity only ( https://www.britannica.com/science/habitable-zone ). The approximation then is for the sqrt(L). For 100,000 this is 316. The habitable would be about 320 AU away from the center of Betelgeuse.
This gives the mind boggling large number 1.3 x 10^6 square astronomical units. The thickness of this material on the scale of proteins and Wolfram Alpha says it is about three times the approximate diameter of a carbon nanotube.
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Hoping to see a Dyson sphere around Betelgeuse would stretch the limits of physics. Or I've got my math off in a few places - which is quite possible.
Edit: I love that you did napkin math for this, BTW.
The place to look for aliens is around Neptune. Solar power (like goldilocks planets, and fission) is for extreme primitives.
No, thanks.
We may want to build a smaller scale parasol for Venus so it cools down and can be eventually settled.