We measure the gas disc thicknesses of the edge-on galaxy NGC 4013 and the less edge-on galaxies ( NGC 4157 and 5907 ) using CO ( CARMA/OVRO ) and/or \ion Hi ( EVLA ) observations .
We also estimate the scale heights of stars and/or the star formation rate ( SFR ) for our sample of five galaxies using Spitzer IR data ( 3.6 \micron and 24 \micron ) .
We derive the average volume densities of the gas and the SFR using the measured scale heights along with radial surface density profiles .
Using the volume density that is more physically relevant to the SFR than the surface density , we investigate the existence of a volumetric star formation law ( SFL ) , how the volumetric SFL is different from the surface-density SFL , and how the gas pressure regulates the SFR based on our galaxy sample .
We find that the volumetric and surface SFLs in terms of the total gas have significantly different slopes , while the volumetric and surface SFLs in terms of the molecular gas do not show any noticeable difference .
The volumetric SFL for the total gas has a flatter power-law slope of 1.26 with a smaller scatter of 0.19 dex compared to the slope ( 2.05 ) and the scatter ( 0.25 dex ) of the surface SFL .
The molecular gas SFLs have similar slopes of 0.78 ( volume density ) and 0.77 ( surface density ) with the same rms scatter .
We show that the interstellar gas pressure is strongly correlated with the SFR but find no significant difference between the correlations based on the volume and surface densities .