Pyramidal truss sandwich cores with relative densities in the range 1 - 10 % have been made from carbon fiber reinforced polymer laminates using a snap-fitting method. The measured quasi-static uniaxial compressive strength increased with increasing from 1 to 11 MPa over the relative density range investigated here. A robust face-sheet/truss joint design was developed to suppress truss - face sheet node fracture. Core failure then occurred by either (i) Euler buckling () or (ii) delamination failure () of the struts. Micro-buckling failure of the struts was not observed in the experiments reported here. Analytical models for the collapse of the composite cores by Euler bucking, delamination failure and micro-buckling of the struts have been developed. Good agreement between the measurements and predictions based on the Euler buckling and delamination failure of the struts is obtained. The micro-buckling analysis indicates this mechanism of failure is not activated until delamination is suppressed. The measurements and predictions reported here indicate that composite cellular materials with a pyramidal micro-structure reside in a gap in the strength versus density material property space, providing new opportunities for lightweight, high strength structural design.