We use a first-passage time approach to study the statistics of the trapping times induced by persistent motion of active particles colliding with flat boundaries. The angular first-passage time distribution and mean first-passage time are calculated exactly for active Brownian and run-and-tumble particles, and the results of the two prototypes of active particles are contrasted. Theoretical predictions are in excellent agreement with Langevin simulations of active particle dynamics. Our results shed further light on how active particles with different dynamics may have equivalent statistical properties in the bulk yet behave differently near boundaries or obstacles.