The size of the contact patch is dictated by the air pressure in the tyre and the weight of the car acting on it. The width of a tyre dictates the operating temperature, basically because a wider tyre has more rubber than a narrower tyre it can dissipate heat more affectively.
I'll try and explain a little better:
Lets take a guess and suggest the front end of a Starlet weighs 550KG, 550KG is approx 1212lbs, so we'll round that down to 1200lbs just to simplify this explanation. OK, so 1200lbs on the front end split between 2 tyres gives us a weight of 600lbs on each tyre.
Next we need our tyre pressure, again trying not to over complicate things we'll pick 30psi as our front tyre pressure. So what does 30psi represent? simply the air pressure in the tyre can support 30 pounds of weight per square inch of surface area. This means that for every 30 pounds of weight sat on the tyre, the contact patch on the ground will increase by 1 square inch.
A simple calculation:
weight / tyre pressure = area of contact patch
i.e. 600lbs / 30psi = 20 square inches
That tells us that both of the tyres on the front end of our Starlet when inflated to 30psi will have a contact patch of 20 square inches. Notice how tyre height or width does not come into this calculation. It doesn't matter what size tyre you fit, the contact patch will always remain the same unless either the weight sat on the tyre is increased/reduced or the air pressure in the tyre is increased/reduced.
By using my awesome Paint skills and without being specific with tyre widths, the difference in the contact patch of a narrow tyre vs a wide tyre fitted to our fictitious Starlet and inflated to the same pressure would look something like this:
(each square represents 1 square inch)
The area of the contact patch of each tyre is the same, only the shape of the contact patch changes.
The amount of 'grip' a tyre can generate is dictated by the
coefficient of friction of the rubber compound it's made from. The force of friction created between the tyre and the road surface generates heat, if the tyre is too narrow (i.e. doesn't have enough rubber to dissipate the heat generated) it'll overheat, if the tyre is too wide the tyre compound won't reach it's optimum operating temperature.
Take F1 cars for example, they use wide tyres. To get the most out of these wide tyres they need to push hard through the corners and accelerate fast down the straight to keep them up to temperature. When the safety car comes out and everyone has to slow down, you see the drivers weaving from one side of the track to the other, if they didn't do this the tyre compound would cool down and it's 'coefficient of friction' would drop. A F1 car entering a corner at high speed with cold tyres will end up in the barrier. If the F1 cars were fitted with narrow tyres to prevent this happening, after a hard lap the tyre would overheat and disintegrate. So there is a element of tyre width tuning going on to give the best tyre temperature control for 'normal' race conditions.
So in essence, the tyre compound and the actual area of the tread pattern in contact with the ground dictates the level of grip achieved.