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We study properties of the so-called inner and outer successive radii of special families of convex bodies. First we consider the balls of the 
$p$-norms, for which we show that the precise value of the outer (inner) radii when 
$p\geq 2$ (
$1\leq p\leq 2$), as well as bounds in the contrary case 
$1\leq p\leq 2$ (
$p\geq 2$), can be obtained as consequences of known results on Gelfand and Kolmogorov numbers of identity operators between finite-dimensional normed spaces. We also prove properties that successive radii satisfy when we restrict to the families of the constant width sets and the 
$p$-tangential bodies.
