This work presents an adaptive method of polygonization for implicit surfaces, the Adaptive Grid, associated to an octree as a data structure. The octree cubes subdivision points are not always set in the center, as in the classical definition, but closer to the surface. In order to obtain this property, the values of the function are considered in the region of polygonization. The mesh points that approximate the surface are connected using the cubes edges related to the leaves in the octree. The edges that belong to the intersection of three or four cubes are tested regarding to its intersection with the surfaces. They are obtained by a redefinition of the Minimal Edges algorithm. The method presented in this work leads to more precise results than the methods for which the octree has its subdivision always in the center, as Dual Contouring. The better positioning of the grid allows to capture more details with the same subdivision level, because in this case there is more accuracy in the mesh vertex positioning and moreover, more cubes intersect the surface, generating more mesh points.