Am I understanding this correctly: All that is left for you to determine the surface area of the cone is to know the area of the base of the cone?

The base of the cone must have a circumference equal to the outer perimeter of the circular cut-out.

First of all, the area of a full disc is , where r is the radius of the disc. Since the central angle of your cut-out is 72°, since the "central angle" of a full disc is 360°, and since 72°/360° = 1/5, the area of your cut-out is given by . You also know that the area of the cut-out is 22 cm^2, whereas you can find the radius r.

By similar reasoning, the outer perimeter of your circular cut-out is , and as you know r, you can find this perimeter. This perimeter is exactly the circumference of the circular base, and once you know the circumference of a disc, you can find its radius, and from the radius, you can find its area.