In this paper, we propose a novel method to extrinsically calibrate a camera to a 3D reference object that is not directly visible from the camera. We use a human cornea as a spherical mirror and calibrate the extrinsic parameters from the reflections of the reference points. The main contribution of this paper is to present a cornea-reflection-based calibration algorithm with a simple configuration: five reference points on a single plane and one mirror pose. In this paper, we derive a linear equation and obtain a closed-form solution of extrinsic calibration by introducing two ideas. The first is to model the cornea as a virtual sphere, which enables us to estimate the center of the cornea sphere from its projection. The second is to use basis vectors to represent the position of the reference points, which enables us to deal with 3D information of reference points compactly. We demonstrate the performance of the proposed method with qualitative and quantitative evaluations using synthesized and real data.