Option 1, an equilateral triangle, is incorrect as it will form a pyramid, not a cone, when it is rotated around one of its edges.

Option 2, a sector of a circle, is incorrect because when it is rotated around one of its straight edges, it will form a portion of a sphere or a spherical sector, not a cone.

Option 3, a segment of a circle, also does not generate a cone when rotated around a straight edge. Instead, it will form a shape known as a spherical cap or dome shape.

Option 4, a right-angled triangle, is the correct answer because when it is rotated about one of its straight edges (specifically the height), it generates a cone. In fact, when a right triangle rotates about one of its legs (the height), it forms a cone where the length of that leg becomes the height of the cone, the other leg becomes the base radius, and the hypotenuse traces the slant height of the cone.