| We present a measurement of the branching fractions of the exclusive radiative penguin processes B → Kpipigamma in a sample of 232 million e+e - → BB¯ decays recorded by the BABAR detector at the PEP-II asymmetric-energy storage ring. We reconstruct four final states: K+pi-pi+gamma, K+pi-pi0gamma, K0s pi-pi+gamma, and K0s pi+pi0gamma, where K0s → pi+pi-, in the range mKpipi < 1.8GeV/c 2. We measure the branching fractions B (B+ → K +pi-pi+gamma) = (2.95+/-0.13(stat.)+/-0.19(syst.)) x 10-5, B (B0 → K +pi-pi0gamma) = (4.07+/-0.22(stat.)+/-0.31(syst.)) x 10-5, B (B0 → K 0pi+pi-gamma) = (1.85+/-0.21(stat.)+/-0.12(syst.)) x 10-5, and B (B+ → K 0pi+pi0gamma) = (4.56+/-0.42(stat.)+/-0.30(syst.)) x 10-5. We also measure the distribution of mKpipi. |