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The edges of the footprint won't be bearing full pressure because they will be in tension (in fact, load distribution profiles of tires tend to look like a rounded trapezoid).

Additionally, the grooves are rather stiff and will help hold the treads - the air pressure on the inside of the grooves will be helping push the treads onto the road. Likewise, tire sidewalls have some stiffness and will also be helping hold the car up beyond just the air pressure, as you've noted in your addendum.

This is a good heuristic for approximating the weight of a car but you won't be getting particularly accurate weight measurements out of it.

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