Here is a picture of the way I would try this problem. I would evaluate the area of the segment, and subtract the area of the triangle. Of course, to do this, you need to know the size of the angle.
I understand your method, but the quoted formula only applies to full circles. It cannot be used to find an area of a segment, sector, etc. Or am I misinterpreting your logic? It seems to me that you are assuming that you can use the '0.5R' as a radius and the given area of the segment of 7.87 to find the radius of the whole circle.