Scintillation NaI(Tl) crystals are typically utilized at room temperature for detection of energetic photons in high energy and nuclear physics research, non-destructive analysis of materials testing, safeguards, verification of nuclear treaty, geological exploration and therapeutic imaging. The present work provides a new geometry for the source-to-detector combination. A special order cubic detector with rectangular cavity was used. The mathematical expressions of the path-lengths traveled by the incident photon as well as the geometrical solid angle were derived. The detector efficiency was determined for an axially positioned standard point-like gamma-ray source using the analytical efficiency transfer technique. Geant4 Monte Carlo simulation code was also used to predict the detector response under the calibration geometry. The analytical efficiency transfer and Geant4 simulation results were compared with those obtained experimentally and a good agreement between them was shown.