Why does metal feel colder than cardboard?

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Because the rate of heat transfer between our hands and the metal is faster than that of the cardboard, thus heat loss of our hands is also faster.

When there's a heat difference of two objects in contact, heat is transferred by conduction.

In metals, they're a good thermal conductor. they have a large amount of free electrons, which makes the heat transfer by conduction the fastest (except diamond), the free electrons travel a long distance before colliding to other molecules.

However, the cardboard lacks of free electrons, plus it has a lot of air spaces, which is very inefficient for heat transfer from our hands to the cardboard, since it traps air, one of the worst thermal conductors.

Therefore, even if both objects are in the same temperature, but because the rate of heat transfer is faster for the metal, our hands feel colder, our hands has a faster heat loss to that.