To prevent wear and damage in the vertex of a nut or a bolt, and to prevent slipping of the nut or bolt in a socket during rotation.

The anti-wearing nut or bolt has polygonal side surface parts on a head periphery of the nut or bolt; and the polygonal side surface part has two inclined sub-side surface parts formed into a small acute angle with a corresponding socket inner side surface part, and an intermediate sub-side surface part intersecting with the two inclined sub side surface parts, respectively inclined to converge toward the corresponding socket inner side surface part of the socket from vertexes in opposing positions of the side surface part. During clockwise or counterclockwise rotation of the socket, one of the inclined sub-side surface parts of the nut or bolt planarly contacts each socket inner side surface part without linearly contacting them, and as a result, wear and damage in the vertex of the nut or bolt are prevented.