If you're looking for exact areas, you need an equal area projection.
But even then they won't agree. This is because the software will assume a straight line in the current projection. For sufficiently densified arcs they will agree.
If you're working with ESRI products, I suppose the closest thing to exact would be Albers Equal Area (based on a cone), because it is the only one that is supported for an ellipsoid. Lambert Azimuthal Equal Area (based on a plane), and Cylindrical Equal Area (based on a cylinder) are supported only for spheres. There are some pseudoconic, pseudocylindrical, and pseudoplanar projections -- I think they are supported only for spheres. There are some equations in one of Snyder's books for ellipsoids in LAEA but they've obviously been too hairy for ESRI to tackle.