The integration of camera and LiDAR technologies has the potential to significantly enhance construction robots’ perception capabilities by providing complementary construction information. Structured light cameras (SLCs) are a desirable alternative as they provide comprehensive information on construction defects. However, fusing these two types of information depends largely on the sensors’ relative positions, which can only be established through extrinsic calibration. This paper introduces a novel calibration algorithm considering a customized board for SLCs and repetitive LiDARs, which are designed to facilitate the automation of construction robots. The calibration board is equipped with four symmetrically distributed hemispheres, whose centers are obtained by fitting the spheres and adoption with the geometric constraints. Subsequently, the spherical centers serve as reference features to estimate the relationship between the sensors. These distinctive features enable our proposed method to only require one calibration board pose and minimize human intervention. We conducted both simulation and real-world experiments to assess the performance of our algorithm. And the results demonstrate that our method exhibits enhanced accuracy and robustness.