Will 185/75R14 tires do the job on a 2001 Toyota Corolla? Stock size is 185/65R14.
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They might fit but your speedometer will be significantly off and you will feel like you've lost a lot of torque.
Tire Rack (as always) has a lot of useful technical reference information on sidewall markings. The sidewall aspect ratio is the measurement in question. That represents the sidewall height as a percentage of the section width (the footprint). So, if we look at both tire sizes:
That's a difference of about 18 mm (or about 3/4 of an inch) at the sidewall. Remember, that's a radial measure: your total tire would now be an additional 36 mm or about 1.5 inches across.
So, to figure out if things fit, you need to inspect your wheel well. Do you have enough clearance for the tires to roll? What if the steering wheel is at its lock: still enough clearance? What if you hit a bump? Is there enough suspension travel that things are going to rub?
Finally, you're going to have to accept that this is going to cause your speedometer to read significantly low. The rolling circumference will go up by something like 20 - 30% so your speedo will be low by about that much. You'll also have to accept that you're giving up a lot of mechanical advantage: the car will feel much slower from a stoplight.
Personally, I wouldn't do this. You may have different goals, though.