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No, it is not true.
pi x radius^2
Because 2Pi x r x L is the curved surface of a cylinder. Clearly a cylinder have more surface area than a cone of same height and radius. The surface of the cone is Pi x r x S where S is the slope length, so the cylinder has approximately double the surface area (note S is longer than L).
The relationship between the surface areas of cylinders, cones, and spheres is that the surface area of a cylinder is equal to the sum of the areas of its two circular bases and its curved surface area, the surface area of a cone is equal to the sum of the area of its circular base and its curved surface area, and the surface area of a sphere is equal to four times the area of its circular base.