Cellular solids such as honeycombs and foams exhibit unusual mechanical properties such as ultra-low density, high strength, and excellent energy absorption capacity. Owing to these distinct properties, they find applications in various engineering fields. Introducing functional gradation of properties in conventional cellular solids has shown to be a promising possibility and has drawn attention over the past few years. In this work, we study the dynamic compressive behavior of foams with gradation in the cross-sectional area. A theoretical model has been developed based on the one-dimensional shock theory and the rigid, perfectly-plastic, and locking (RPPL) material model, to establish the governing equations of motion of an initially stationary foam struck by a rigid projectile. Both decreasing and increasing cross-sectional area profiles with various power-law exponents are considered. The numerical simulations suggest that the decreasing area profiles, referred to as the negative area-graded foams, exhibit a double-shock mode. In contrast, the foams with an increasing area-profile, referred to as the positive area-graded foams, exhibit a single-shock mode. The theoretical predictions of both the double-shock and the single-shock cases are validated against the finite element simulations. Compared to the foam with a uniform cross-sectional area, the negative area-graded foams transmit lower mean forces at the distal end, and the positive area-graded foams transmit higher forces at higher impact velocities. The mean force at the distal end decreases with an increase in the impact velocity beyond the densification velocity, for the negative area-graded foams. The performance of the foams has been compared through the dissipation performance parameter. It is found that the negative area-graded foams with low gradients and high power-law exponents have the highest dissipation performance. A special case of a negative density-gradient combined with a positive area-gradient is shown (through finite element simulations) to be beneficial for structural protection.