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I agree, the sanity check is not sufficient. It confirms the concept that as pressure decreases, area increases, but is not sufficient to confirm a linear relationship. In the limit, the area clearly does not become infinite when pressure drops to 0.

For the case you suggested above, this is essentially how run-flat tires work. Of course, the sidewall of the tire isn't as stiff as the support ring, but they do provide some support.

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