Imagine a spherical target (shown as the dashed grey and red circle in the figure) and a beam of particles (in blue) "flying" at speed v (vector in blue) in the direction of the target. We want to know how many particles impact it during time interval d t. To achieve it, the particles have to be in the green cylinder in the figure (volume V). The base of the cylinder is the geometrical cross section of the target perpendicular to the beam (surface σ in red) and its height the length travelled by the particles during d t (length v d t):