In this paper, we estimate the number of gravitational arcs detectable in a wide-field survey such as that which will be operated by the Euclid space mission, assuming a Λ cold dark matter cosmology. We use the publicly available code moka to obtain realistic deflection angle maps of mock gravitational lenses. The maps are processed by a ray-tracing code to estimate the strong lensing cross-sections of each lens. Our procedure involves (1) the generation of a light-cone which is populated with lenses drawn from a theoretical mass function, (2) the modelling of each single lens using a triaxial halo with a Navarro–Frenk–White density profile and theoretical concentration–mass relation, including substructures, (3) the determination of the lensing cross-section as a function of redshift for each lens in the light-cone and (4) the simulation of mock observations to characterize the redshift distribution of sources that will be detectable in the Euclid images. We focus on the so-called giant arcs, i.e. gravitational arcs characterized by large length-to-width ratios (l/w > 5, 7.5 and 10). We quantify the arc detectability at different significances above the level of the background. Performing 128 different realizations of a 15 000 deg2 survey, we find that the number of arcs detectable at 1σ above the local background will be , and for l/w ≥ 5, 7.5 and 10, respectively. The expected arc numbers decrease to , and for a detection limit at 3σ above the background level. From our analysis, we find that most of the lenses which contribute to the lensing optical depth are located at redshifts 0.4 < zl < 0.7 and that the 50 per cent of the arcs are images of sources at zs > 3. This is the first step towards the full characterization of the population of strong lenses that will be observed by Euclid. Given these results, we conclude that Euclid is a powerful instrument for strong lensing related science, which will be useful for several applications, ranging from arc and Einstein ring statistics to the measurement of the matter content in the cluster cores.