A method is proposed to measure the photon polarisation parameter $$\lambda _{\gamma }$$ ${\lambda }_{\gamma }$ in $${{b}} \!\rightarrow {s} {\gamma }$$ $b\to s\gamma$ transitions using an amplitude analysis of $$B\!\rightarrow K\pi \pi {\gamma }$$ $B\to K\pi \pi \gamma$ decays. Simplified models of the $${K} {\pi } {\pi }$$ $K\pi \pi$ system are used to simulate $${{{B} ^+}} \!\rightarrow {{K} ^+} {{\pi } ^-} {{\pi } ^+} {\gamma }$$ $B^{+}\to K^{+}{\pi }^{-}{\pi }^{+}\gamma$ and $${{B} ^0} \!\rightarrow {{K} ^+} {{\pi } ^-} {{\pi } ^0} {\gamma }$$ $B^0\to K^{+}{\pi }^{-}{\pi }^0\gamma$ decays, validate the amplitude analysis method, and demonstrate the feasibility of a measurement of the $$\lambda _{\gamma }$$ ${\lambda }_{\gamma }$ parameter irrespective of the model parameters. Similar sensitivities to $$\lambda _{\gamma }$$ ${\lambda }_{\gamma }$ are obtained with both the charged and neutral hadronic systems. In the absence of any background and distortion due to experimental effects, the statistical uncertainty expected from an analysis of $${{{B} ^+}} \!\rightarrow {{K} ^+} {{\pi } ^-} {{\pi } ^+} {\gamma }$$ $B^{+}\to K^{+}{\pi }^{-}{\pi }^{+}\gamma$ decays in an LHCb data set corresponding to an integrated luminosity of 9 $$\,\hbox {fb}^{-1}$$ ${\text{fb}}^{-1}$ is estimated to be 0.009. A similar measurement using $${{B} ^0} \!\rightarrow {{K} ^+} {{\pi } ^-} {{\pi } ^0} {\gamma }$$ $B^0\to K^{+}{\pi }^{-}{\pi }^0\gamma$ decays in a Belle II data sample corresponding to an integrated luminosity of 5 $$\hbox {\,ab}^{-1}$$ ${\text{ab}}^{-1}$ would lead to a statistical uncertainty of 0.018.