Describe the theory associated with the radius of an atom as it:
$(a)$ Gains an electron
$(b)$ Loses an electron

(N/A) The atomic radius is defined based on the nature of the bond,such as metallic radius or covalent radius,which is half the internuclear distance between bonded atoms.
$(a)$ When an atom gains an electron,it forms a negative ion (anion). The radius of an anion is always larger than that of its parent neutral atom. This is because the addition of one or more electrons increases the electron-electron repulsion,while the effective nuclear charge $(Z_{eff})$ decreases,causing the electron cloud to expand.
$(b)$ When an atom loses an electron,it forms a positive ion (cation). The radius of a cation is always smaller than that of its parent neutral atom. This is because the loss of electrons results in a higher effective nuclear charge per remaining electron,pulling the electron cloud closer to the nucleus.
In summary,for isoelectronic species: $\text{radius of cation} < \text{radius of neutral atom} < \text{radius of anion}$.