Fiber-reinforced polymer (FRP) has been used for seismic retrofitting and structural reinforcement over recent decades. Numerous researchers have created stress–strain models based on experimental data to predict the mechanical properties of FRP-confined concrete. In this study, circular and square cross-section specimens with different design concrete strength were prepared, and the compressive strength of the specimens confined with different layers of aramid FRP (AFRP) were measured in compressive tests. A constitutive model was proposed to simulate the uniaxial compressive stress–strain relationship of the AFRP-confined concrete, which was derived from the Mohr–Coulomb failure envelope theory, and the corresponding axial strain was determined from the regression analysis. The internal friction angle of the proposed constitutive model was determined for the cylindrical concrete specimens confined with one and two layers of AFRP. The compressive strength of one and two layers of AFRP-confined concrete specimens were used to obtain the parameters of the constitutive model; the absolute average error between experimental and predicted compressive strength was 7.01%. Then, the constitutive model was used to predict the strength of a three-layer AFRP-confined concrete specimen, and the absolute average error was 4.95%. The cross-sectional shape coefficient of the square concrete specimen was obtained analytically. Substituting the cross-sectional shape coefficient into the proposed constitutive model, the average absolute error of the square cross-section concrete specimen was about 3.84%. The results indicated that the proposed constitutive model can predict the compressive strength of circular and square cross-section concrete specimens confined with AFRP.