It has been suggested that planetary radii increase with the stellar mass, for planets below 6 R⊕ and host below 1 M⊙. In this study, we explore whether this inferred relation between planetary size and the host star’s mass can be explained by a larger planetary mass among planets orbiting more massive stars, inflation of the planetary radius due to the difference in stellar irradiation, or different planetary compositions and structures. Using exoplanetary data of planets with measured masses and radii, we investigate the relations between stellar mass and various planetary properties for G- and K- stars, and confirm that more massive stars host larger planets and more massive. We find that the differences in the planetary masses and temperatures are insufficient to explain the measured differences in radii between planets surrounding different stellar types. We show that the larger planetary radii can be explained by a larger fraction of volatile material (H-He atmospheres) among planets surrounding more massive stars. We conclude that planets around more massive stars are larger most probably as a result of larger H-He atmospheres. Our findings imply that planets forming around more massive stars tend to accrete H-He atmospheres more efficiently.