The transiting exoplanetary system WASP-174 was reported to be composed by a main-sequence F star (V=11.8 mag) and a giant planet, WASP-174b (orbital period 4.23 days). However only an upper limit was placed on the planet mass (<1.3 Mj), and a highly uncertain planetary radius (0.7-1.7 Rj) was determined. We aim to better characterise both the star and the planet and precisely measure their orbital and physical parameters. In order to constrain the mass of the planet, we obtained new measurements of the radial velocity of the star and joined them with those from the discovery paper. Photometric data from the HATSouth survey and new multi-band, high-quality (precision reached up to 0.37~mmag) photometric follow-up observations of transit events were acquired and analysed for getting accurate photometric parameters. We fit the model to all the observations, including data from the TESS space telescope, in two different modes: incorporating the stellar isochrones into the fit, and using an empirical method to get the stellar parameters. The two modes resulted to be consistent with each other to within 2 sigma. We confirm the grazing nature of the WASP-174b transits with a confidence level greater than 5 sigma, which is also corroborated by simultaneously observing the transit through four optical bands and noting how the transit depth changes due to the limb-darkening effect. We estimate that ~76% of the disk of the planet actually eclipses the parent star at mid-transit of its transit events. We find that WASP-174b is a highly-inflated hot giant planet with a mass of 0.330 Mj and a radius of 1.435 Rj, and is therefore a good target for transmission-spectroscopy observations. With a density of 0.135 g/cm^3, it is amongst the lowest-density planets ever discovered with precisely measured mass and radius.
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