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Ionic radii of cations:

When you form a metal cation, you are removing the outermost electrons and revealing the inner core.  

For example, to form an Na⁺ ion you must remove a 3s electron from an Na atom, exposing the 2s and 2p subshells.

The n = 2 shell is smaller than an n= 3 shell, so an Na⁺ ion is smaller than an Na atom (as in the image below).

Ionic radii of anions:

When you form an anion, you are adding electrons to the outermost shell. These electrons (a) increase the shielding from the nucleus and (b) repel all the other electrons.  

The decreased nuclear attraction and the electronic repulsions cause the electron cloud to expand, so a Cl⁻ ion is larger than a Cl atom.