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This method doesn't account for the support of the tire itself through the sidewalls, as opposed to the pressure contained within it.

You should be able to estimate this contribution by fitting to the curve of area vs pressure at lower pressures. For example, the linear F=PA relationship clearly breaks down at 0 pressure, as F>0 and A>0 when P=0.

Except (as mentioned further up this post) when the side walls are in tension due to the air pressure within the tyre.

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