Circle detection in digital images is an important problem in computer vision, pattern recognition, and artificial intelligence. However, the common circle detection strategies, including random sample consensus, randomized Hough transform, and randomized circle detection, have a very low sampling efficiency and thus a slow detection speed, owing to aimless random sampling. This paper proposes a fast and accurate randomized circle detection algorithm, with the aim to improve the speed and accuracy of circle detection based on random sampling. The proposed algorithm mainly focuses on four aspects: calculating circle parameters, determining candidate circles, searching for true circle, and improving detection accuracy. To verify the effectiveness of our algorithm, contrastive experiments were conducted on lots of synthetic and real images. The results show that our algorithm achieved much higher detection speed and accuracy than random sample consensus, randomized Hough transform, and randomized circle detection, and realized similar robustness as the three contrastive strategies. The research ideas can also be applied to ellipse detection.