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An atomic radius is defined as half the distance between the centers of neighboring atoms. This inclues both ionic and covalent radii as well. A covalent radius is specifically referring to two atoms joined by a chemical bond while an ionic radius refers to the distance between neighboring ions in an ionic solid.
One difference between atomic and covalent radii versus ionic radii is that when we measure an ionic radius, it's more complicated than simply dividing the distance between the nuclei of two atoms in the ionic bond because they are of different size. Also, the ionic radius of, for example, chlorine, will be different from its atomic radius because it has another electron and it will be larger.
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