Epitaxial ultrathin Fe films on fcc Cu(001) exhibit a spin spiral (SS), in contrast to the ferromagnetism of bulk bcc Fe. We study the in-plane (IP) and out-of-plane (OP) Fermi surfaces (FSs) of the SS in 8 monolayer Fe/Cu(001) films using energy-dependent soft-x-ray momentum-resolved photoemission spectroscopy. We show that the SS originates in nested regions confined to OP FSs, which are drastically modified compared to IP FSs. From precise reciprocal-space maps in successive zones, we obtain the associated real space compressive strain of 1.5+/-0.5% along c axis. An autocorrelation analysis quantifies the incommensurate ordering vector q=(2pi/a)(0,0, approximately 0.86), favoring a SS and consistent with magneto-optic Kerr effect experiments. The results reveal the importance of IP and OP FS mapping for ultrathin films.