This letter investigates the optimal hovering altitude of a single UAV toprovide coverage over any convex quadrilateral region on the ground. The UAVemploys a directional antenna with a tiltable beam, producing an ellipticalcoverage pattern. Two scenarios are considered: (1) inscribing the largestellipse within the quadrilateral to cover its interior, and (2) circumscribingthe smallest ellipse about the quadrilateral to ensure full coverage. We derivethe optimal UAV altitude and antenna tilt conditions in both scenarios for asimplified yet widely accepted path loss model and present numerical resultsfor coverage efficiency. The work contributes to the development ofenergy-efficient UAV-based communication systems.