Here's a way to get the area of the slant surface of the cone. Imagine cutting a slit up the side, then flatten it out (which, unlike the surface of a sphere, can be done). You get a part of a circle. The radius of the circle is equal to the length of the slant side (which you can get by the Pythagorean theorem with height and radius of the cone as legs). The circumference of the base of the cone will NOT be which is why you do not have a full circle. But the area of this flattened cone will be the same fraction of the area of the full circle as its circumference is of the circumference of the full circle.