Probabilistic clustering algorithms extend network lifetimes by rotating cluster heads among nodes; however, they face significant challenges such as determining the optimal number of cluster heads, managing the geographic distribution of cluster heads, and balancing energy consumption. The proposed CAT algorithm addresses these issues by determining the exact required number of cluster heads, achieving a well-distributed placement of cluster heads without relying on complex localization techniques, and reducing intra-cluster energy consumption to balance overall energy use. CAT is designed based on the area tessellation theorem and offers a novel method for selecting the optimal number of cluster heads. The algorithm was tested on various network dimensions, including a common deployment scenario from the literature. Results show that CAT outperforms prominent clustering algorithms such as LEACH, HEED, DDR, and ELDCA. Moreover, CAT extends the network lifetime when the network radius exceeds the crossover distance of the free-space model. Simulations indicate that CAT can extend network lifetimes by 21% to 50% while maintaining near-linear node energy consumption.