What are the trade-offs in gas mileage & performance?

My car has 12 inch wheels. I have a general question about tire/wheel size — diameter — and how it affects gas mileage.
Why do cars like mine come with 12 inch wheels and tires?
It seems as if larger diameter wheels and tires would get better mileage at the risk of a little loss in performance.
Is there a trade-off point where under-powered motors won’t handle those larger sizes?
Would it be a mistake to replace wheels/tires on a small car with a larger diameter?

The primary reason this type of vehicle has smaller sized wheels is to make it practical for its engine to effectively power the car.
The engineers used design specifications and costs to match the performance, gasoline mileage, and safety needs to the vehicle’s weight and anticipated uses.
If the wheels are changed to a larger diameter, the leverage applied by the engine to twist the wheels — thus moving the car forward — will be less than the designers deemed safe and practical.
If the wheels and tires were to be increased in diameter there would be an inordinate increase in unsprung weight which would negatively impact the car’s ride and handling.
Also, gasoline mileage might theoretically increase by a relatively small amount, but it is likely to be unnoticeable — or even decrease — due to the increased load placed upon the engine. If it has a manual shift transmission, you will likely feel the need to over-rev the engine in order to start up from a standing start. You will likely need to shift into second and third gears at higher RPMs. These operational driving aspects will contribute to an increase in gasoline usage.
To identify the exact trade-off point where mileage, power, and other factors cross and become cost-effective and remain practical would require very controlled testing changing equipment and recording results for subjective analysis. That testing is what the automobile manufacturers’ engineers have done for you. They determined that that car with that engine and anticipated driving needs would be best using the wheels supplied.
An exception would be if the manufacturer offered an upgraded wheel/tire combination for the car when new.
If there was a wheel/tire optional upgrade, you can consider making that change. However, considering the cost of new wheels and tires it would require many miles to recoup that expense in gasoline savings.

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