We present a method of determining the number of contacts in a planar fibrous network by first determining its average fibre segment length. An analytical expression is derived for the mean fibre segment length in terms of the average shortest path along fibres through the sample in the out-of-plane direction. The model is validated by applying the method to generated fibre networks, for which the segment lengths are known. Both the theoretical and experimental results show that the contact density is essentially proportional to the square of the solid content with a prefactor that depends only on the geometry of the fibres. The method is also tested on physical samples of cardboard, which were imaged with high resolution micro X-ray computed tomography. The results of the cardboards show similar behaviour as the generated fibre networks.