Which has a larger atomic radius, potassium or bromine, and why?

Both Bromine with an atomic number of 35 and Potassium with an atomic number of 19 both belong to group 4.  The 'rule' is that the atomic radius increases across the group from Group One to Eight.  This can be proven true.

As you move across the a period from left to right, the amount shell remains the same however an electron is added on to the final orbital.  In the case of potassium which has only one shell on the final shell, that electron is shielded from the pull of the nucleus by the third shell of electrons and as such the shell is able to float further away from the rest of the shells thus making a larger atom on a whole.  In the case of Bromine the number of electrons it has on the fourth shell (one down from the octet) is pulled heavily toward the nucleus because the increase in the number of protons in the nucleus thus seeing where its shells are closer to the nucleus therefore contributing to a smaller atomic radius.

Just so that you will always remember this here is a picture to remind you always.