Many theoretical models of high-temperature superconductivity focus only on the doping dependence of the CuO(2)-plane electronic structure. However, such models are manifestly insufficient to explain the strong variations in superconducting critical temperature, T(c), among cuprates that have identical hole density but are crystallographically different outside of the CuO(2) plane. A key challenge, therefore, has been to identify a predominant out-of-plane influence controlling the superconductivity, with much attention focusing on the distance d(A) between the apical oxygen and the planar copper atom. Here we report direct determination of how variations in interatomic distances within individual crystalline unit cells affect the superconducting energy-gap maximum Delta of Bi(2)Sr(2)CaCu(2)O(8+delta). In this material, quasiperiodic variations of unit cell geometry occur in the form of a bulk crystalline "supermodulation." Within each supermodulation period, we find approximately 9 +/- 1% cosinusoidal variation in local Delta that is anticorrelated with the associated d(A) variations. Furthermore, we show that phenomenological consistency would exist between these effects and the random Delta variations found near dopant atoms if the primary effect of the interstitial dopant atom is to displace the apical oxygen so as to diminish d(A) or tilt the CuO(5) pyramid. Thus, we reveal a strong, nonrandom out-of-plane effect on cuprate superconductivity at atomic scale.