Assuming using all the mass in the solar system (1.82 10^26 kg), and 1AU, you get something that is about 8-20 cm thick depending on density.

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.

Just for the record: the Dyson sphere is not an actual solid hollow sphere around a star. Rather it is a swarm of rotating objects orbiting the star.

You're thinking of a Dyson swarm.

The naming is kinda misleading as Freeman Dyson envisioned the "Dyson sphere" as a swarm of objects. Although "sphere" is not technically correct, both swarm and sphere refer to the same thing here: a swarm of objects (e.g., O'Neill cylinders) orbiting a star.

Huh, TIL. I always thought they were two separate ideas, not that the Dyson swarm is a type of Dyson sphere.

Why would a hypothetical Dyson sphere need to be in the habitable zone? I would guess that it should be built as small as the construction materials can tolerate.

Edit: I love that you did napkin math for this, BTW.

We can confidently predict that no one will ever find a Dyson sphere of any description. Anybody equipped to make one would have long since perfected controlled fusion, and spent the time fusing the nearest cold gas giant after mining out the actually frozen ones.

The place to look for aliens is around Neptune. Solar power (like goldilocks planets, and fission) is for extreme primitives.

An actual sphere is implausible and probably impractical. (Unstable, no gravity, no atmosphere). There doesn’t seem to be any reason to orbit artificial structures in the habitable zone.

Around a star that's going supernova in the next 1e+5 years?

No, thanks.

maybe it's stadium seating for a species that doesn't mind a few gamma rays

Or an alien doomsday cult.

Or a shield to protect things nearby from the coming supernova.

A "paranova" is an interesting concept, but it'd probably be more effective to build multiple smaller ones closer to the settled worlds.

We may want to build a smaller scale parasol for Venus so it cools down and can be eventually settled.

IIRC this dimming cycle has been repeating for decades

Actually there is several period detected in the brightness, one being ~420 days (give or take 10 days or so), another is ~2100 days. Several of those having a (nearly) simultaneous minimum might very well explain the stronger than normal dimming we just have seen. The other thing that got some attention is that Betelgeuse did not start to rebrighten January twenty-something -- like expected -- but reached a minimum slightly later, on February 10th or so. The correct conclusion of course is not "it is blowing up" but "we got the period length slightly wrong", closer to 425 days or maybe the period length is not exactly constant.