= Solution
The two relevant categories are \b[strongly irradiated, often transiting <hot Jupiters>, and young <directly imaged exoplanets> with large residual-entropy radii]. The young objects are inflated relative to an old cold mass-radius sequence; a large young radius is not automatically anomalous relative to an age-appropriate formation model.
For a transit, the depth gives $(R_p/R_\star)^2$ after geometric and <limb darkening> corrections; an independently estimated stellar radius sets $R_p$. A measured planet mass then tests its location on the <planetary mass-radius relation>. For a directly imaged object, distance and integrated spectral flux give luminosity; a fitted <effective temperature> supplies
$$
\boxed{R_p=\left(\frac{L}{4\pi\sigma_{\rm SB}T_{\rm eff}^4}\right)^{1/2}.}
$$
The imaged radius is model-dependent because temperature, gravity and <exoplanet clouds> are inferred from its spectrum; its mass may also depend on age and evolutionary models unless dynamically measured.
Two broad explanatory classes are \b[slower loss of existing heat and addition of heat to the deep interior]. Specific delayed-cooling proposals are enhanced atmospheric <opacity>, which restricts radiative loss, and composition gradients producing <layered convection in a giant planet>. Specific heating proposals are <tidal heating> and <Ohmic heating> from currents induced by magnetized atmospheric winds, the proposed <Ohmic heating of a giant planet>. Heating must reach or influence sufficiently deep layers to maintain the interior <specific entropy>; merely heating the optically thin upper atmosphere is not equivalent. For young distant objects, retention of formation heat, described by <hot and cold starts of a giant planet>, itself explains much of the large radius without requiring the hot-Jupiter heating mechanisms.
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