Extremities have a high surface area to volume ratio, meaning there is less ‘stuff’ underneath every square centimeter of skin. Your skin radiates heat and the stuff underneath generates that heat so you lose a lot more heat from hands and feet than you do elsewhere.
Hands and feet also have high concentrations of sensory nerves so you feel things more intensely in those areas and will feel cold or hot there first.
So you lose heat most rapidly from the places that are most likely to notice a temperature difference.
This is actually something of a mistaken assumption that gets the logic backwards.
Scientists have done studies that show that people lose the most heat through hands, feet, and head in very cold situations *while dressed for the cold*. They take a thermal image, which shows the most heat around those areas. And a lot of people have interpreted this to mean that those areas lose the most heat, which causes this. But the actuality is that people lose the most heat through those areas because it is harder to extensively cover them while still maintaining enough functionality to do anything. Your core/torso is actually the place where you would lose the most heat if it’s exposed, but it’s very easy to layer up your torso with multiple layers of clothing, insulating it well. Meanwhile, you sacrifice significant dexterity in your hands by wearing even one pair of relatively thin gloves, and going beyond that rapidly diminishes utility. Likewise, your feet have to fit into your shoes/boots so you can’t just wear six pairs of socks, and it’s difficult to fully shield the face from cold exposure without also blocking your vision. There also tend to be more gaps, i.e. between your sleeves and your gloves, between your pants and your shoes, and between your collar and your head covering, which gives an avenue for heat to escape.
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