In this work we theoretically contribute to neural network approximation by providing a novel tropical geometrical viewpoint to structured neural network compression. In particular, we show that the approximation error between two neural networks with ReLU activations and one hidden layer depends on the Hausdorff distance of the tropical zonotopes of the networks. This theorem comes as a first step towards a purely geometrical interpretation of neural network approximation. Based on this theoretical contribution, we propose geometrical methods that employ the K-means algorithm to compress the fully connected parts of ReLU activated deep neural networks. We analyze the error bounds of our algorithms theoretically based on our approximation theorem and evaluate them empirically on neural network compression. Our experiments follow a proof-of-concept strategy and indicate that our geometrical tools achieve improved performance over relevant tropical geometry techniques and can be competitive against non-tropical methods.