Much of the stuff we discuss about tires turns out to be completely counterintuitive.
Here are some mind-altering considerations which just might make your head hurt:

The *area* of the contact patch is based not on tire size but on, ahem, inflation pressure.
Read that twice, and think for a moment.

If your air pressure is holding the car up, then the contact area is based on that: If you're running, say, 30 pounds per square inch, and you've got 1200 pounds on the wheel, then your contact patch is 1200/30 = 40 square inches. In reality it'll be a bit less because of the tread pattern and such but that's the general idea.

The first weird thing you'll notice about that: increasing your tire pressure *reduces* your contact area. If you want a bigger contact patch, you should let some air out of your tires...?

Why, then, do we want higher tire pressure when we're driving hard?
Because traction is not based on area.

Huh?
Yes, really. Traction is the coeff. of friction times the downforce.
Note that area is not part of that equation. Look it up. I'll be here when you get back.

Now, repeat three times: increasing the area of contact does not increase the traction.

Okay, now that your brain has melted, let's forget all that stuff about area, because even a physicist who is very familiar with this contact area stuff, will admit that if you're going racing you're gonna want big tires.

Why?

We know that tires deform under load. After all, that's why we have tires: to conform to the road surface. The taller the tire and/or the lower the inflation pressure, the greater the deformation. High pressure, less deformation. Low pressure, smooth ride.

But: once the tire deforms significantly in a corner, things get kinda bad - the tread rolls underneath, you start scrubbing the sidewalls, etc. That's why things get hairy at the limit on narrow or underinflated tires: the tire is deforming in a non-copacetic way. The tire is no longer particularly round, nor very perpendicular to the road surface. (underinflation also makes hydroplaning worse, but we won't go into that here.)

Anyway, we've gone to wide tires not because of area, but because they keep their shape better than tall ones do. Even though the contact patch is the same size (not that it matters) the behavior under side loads is a whole lot better and more predictable. The contact patch stays out at the corners of the vehicle where it belongs, and everything remains closer to the correct shape so it doesn't start to scrub as you turn. If you like corners, get wide low-profile tires. Notice how the tires with the big tread blocks do best in dry conditions, and slicks are the best of all? It's not about area; it's about deformation.

Tall tires deform more easily, which makes handling more wiggly in corners but gives you a nice cushy ride. Like all things, it's a trade-off.

Ain't physics grand?