We have analyzed the Barrow holographic dark energy (BHDE) in the framework of the flat FLRW Universe by considering the various estimations of Barrow exponent $triangle$. Here we define BHDE, by applying the usual holographic principle at a cosmological system, for utilizing the Barrow entropy rather than the standard Bekenstein-Hawking. To understand the recent accelerated expansion of the universe, considering the Hubble horizon as the IR cut-off. The cosmological parameters, especially the density parameter ($Omega_{_D}$), the equation of the state parameter ($omega_{_D}$), energy density ($rho_{_{D}}$) and the deceleration parameter($q$) are studied in this manuscript and found the satisfactory behaviors. Moreover, we additionally focus on the two geometric diagnostics, the statefinder $(r,s)$ and $O_{m}(z)$ to discriminant BHDE model from the $Lambda CDM$ model. Here we determined and plotted the trajectories of evolution for statefinder $(r, s)$, $(r,q)$ and $O_{m}(z)$ diagnostic plane to understand the geometrical behavior of the BHDE model by utilizing Planck 2018 observational information. Finally, we have explored the new Barrow exponent $triangle$, which strongly affects the dark energy equation of state that can lead it to lie in the quintessence regime, phantom regime, and exhibits the phantom-divide line during the cosmological evolution.
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