Formal verification of neural networks is essential for their deployment in safety-critical real-world applications, such as autonomous driving and cyber-physical controlling. Multi-neuron convex relaxation is one of the mainstream methods to improve verification precision. However, existing techniques rely on empirically selecting neuron groups before performing multi-neuron convex relaxation, which may yield redundant yet expensive convex hull computations. This paper proposes a volume approximation-based approach for selecting neuron groups. We approximate the volumes of convex hulls for all group candidates, without calculating their convex hulls. The group candidates with small volumes are then selected for convex hull computation, aiming at ruling out unnecessary convex hulls with loose relaxation. We implement our approach as the neural network verification tool FaGMR, and evaluate it with state-of-the-art tools on neural networks trained by MNIST and CIFAR-10. The experimental results demonstrate that FaGMR is more efficient than the state-of-the-art works, yet with better precision in most of the cases.