A geometric model of a physical network affected by a disaster is proposed and analyzed using integral geometry (geometric probability). This analysis provides a theoretical method of evaluating performance metrics, such as the probability of maintaining connectivity, and a network design rule that can make the network robust against disasters. The proposed model is of when the disaster area is much larger than the part of the network in which we are interested. Performance metrics, such as the probability of maintaining connectivity, are explicitly given by linear functions of the perimeter length of convex hulls determined by physical routes. The derived network design rule includes the following. (1) Reducing the convex hull of the physical route reduces the expected number of nodes that cannot connect to the destination. (2) The probability of maintaining the connectivity of two nodes on a loop cannot be changed by changing the physical route of that loop. (3) The effect of introducing a loop is identical to that of a single physical route implemented by the straight-line route.