A cutting insert of an almost polygonal shaped plate that is to be mounted detachably on a tool body of an indexable face mill, comprising: a rake surface that is provided for the top face of the polygonal shape; a seat surface that is provided for the bottom face, opposite the top face; main flank surfaces that are substantially flat surfaces having positive relief angles, provided on side faces extended from side ridges of the top face; pairs of sub-flank surfaces that are provided on side faces extended from vertex angle portions of the polygonal shaped faces; and vertex flank surfaces that are provided at the vertex angle portions between the pairs of sub-flank surfaces, and are formed in a substantially projected curved shape, wherein the vertex flank surfaces are smoothly connected with the sub-flank surfaces in a tangential direction, and are extended along a thickness of the cutting insert in a direction along a vertex angle bisector line; characterized in that boundary lines, formed by the vertex flank surfaces and the pairs of sub-flank surfaces, are parallel to each other; wherein as viewed from a direction opposite the rake surface, almost the same shape is provided for curved ridges, formed along an intersecting ridge line portion of the vertex flank surfaces and the rake surface, and for curved ridges that are formed along an intersecting ridge line portion of the vertex flank surfaces and the seat surface; wherein the vertex flank surfaces are extended from the rake surface to the seat surface with a constant radius of curvature being maintained; and wherein relief angles in cross sections taken along a normal line of the respective curved ridge are gradually reduced in a direction leading from an intersecting point of the curved ridge and the bisector line to the two ends of the curved ridge. a rake surface that is provided for the top face of the polygonal shape; a seat surface that is provided for the bottom face, opposite the top face; main flank surfaces that are substantially flat surfaces having positive relief angles, provided on side faces extended from side ridges of the top face; pairs of sub-flank surfaces that are provided on side faces extended from vertex angle portions of the polygonal shaped faces; and vertex flank surfaces that are provided at the vertex angle portions between the pairs of sub-flank surfaces, and are formed in a substantially projected curved shape, wherein the vertex flank surfaces are smoothly connected with the sub-flank surfaces in a tangential direction, and are extended along a thickness of the cutting insert in a direction along a vertex angle bisector line; characterized in that boundary lines, formed by the vertex flank surfaces and the pairs of sub-flank surfaces, are parallel to each other; wherein as viewed from a direction opposite the rake surface, almost the same shape is provided for curved ridges, formed along an intersecting ridge line portion of the vertex flank surfaces and the rake surface, and for curved ridges that are formed along an intersecting ridge line portion of the vertex flank surfaces and the seat surface; wherein the vertex flank surfaces are extended from the rake surface to the seat surface with a constant radius of curvature being maintained; and wherein relief angles in cross sections taken along a normal line of the respective curved ridge are gradually reduced in a direction leading from an intersecting point of the curved ridge and the bisector line to the two ends of the curved ridge.