AbstractA graph is 1‐planar if it can be drawn in the plane such that each edge is crossed at most once. A graph, together with a 1‐planar drawing is called 1‐plane. A graph is maximal 1‐planar (1‐plane), if we cannot add any missing edge so that the resulting graph is still 1‐planar (1‐plane). Brandenburg et al. showed that there are maximal 1‐planar graphs with only edges and maximal 1‐plane graphs with only edges. On the other hand, they showed that a maximal 1‐planar graph has at least edges, and a maximal 1‐plane graph has at least edges. We improve both lower bounds to .