We present a measurement of the partial branching fractions and mass spectra of the exclusive radiative penguin processes B → K π π γ in the range m K π π < 1.8     GeV / c 2 . We reconstruct four final states: K + π − π + γ , K + π − π 0 γ , K 0 S π − π + γ , and K 0 S π + π 0 γ , where K 0 S → π + π − . Using 232 × 10 6 e + e − → B ¯¯¯ B events recorded by the BABAR experiment at the SLAC PEP-II asymmetric-energy storage ring, we measure the branching fractions B ( B + → K + π − π + γ ) = [ 2.95 ± 0.13 ( stat ) ± 0.20 ( syst ) ] × 10 − 5 , B ( B 0 → K + π − π 0 γ ) = [ 4.07 ± 0.22 ( stat ) ± 0.31 ( syst ) ] × 10 − 5 , B ( B 0 → K 0 π + π − γ ) = [ 1.85 ± 0.21 ( stat ) ± 0.12 ( syst ) ] × 10 − 5 , and B ( B + → K 0 π + π 0 γ ) = [ 4.56 ± 0.42 ( stat ) ± 0.31 ( syst ) ] × 10 − 5 .