An island is in the shape of a convex polygon with perimeter \(p\) miles. Its territorial water consists of all points of the sea within \(r\) miles of its coast. Assuming that no other land is within \(r\) miles of the island and that the earth is flat, prove that the area of the territorial water is \((pr+\pi r^2)\) square miles.

Consider whether the same formula would hold

  1. if the island were convex but not polygonal,

  2. if it were polygonal but not convex.