Using the Hilbert-Huang transform technique, we investigate the midrange periodicities in solar radio flux at 2800 MHz (F10.7) and sunspot areas (SAs) from February 1, 1947 to September 30, 2016. The following prominent results are found: (1) The quasi-periodic oscillations of both data sets are not identical, such as the rotational cycle, the midrange periodicities, and the Schwabe cycle. In particular, the midrange periodicities ranging from 37.9 days to 297.3 days are related to the magnetic Rossby-type waves; (2) The 1.3-year and 1.7-year fluctuations in solar activity indicators are surface manifestations (from photosphere to corona) of magnetic flux changes generated deep inside the Sun; (3) At the timescale of the Schwabe cycle, the complicated phase relationships in the three intervals (1947-1958, 1959-1988, and 1989-2016) agree with the produced periodicities of the magnetic Rossby-type waves. The findings indicate that the magnetic Rossby-type waves are the possible physical mechanism behind the midrange periodicities of solar activity indicators. Moreover, the significant change in the relationship between photospheric and coronal activity took place after the maximum of solar cycle 22 could be interpreted by the magnetic Rossby-type waves.