The 40 segments on the boundary each contribute to the perimeter of one polygon, and the 80 segments inside the grid each contribute to the perimeter of two polygons. So the total perimeter of the 20 polygons is 200. Since each polygon has equal area and consists of grid squares, each polygon is a pentomino, and the only pentomino with perimeter at most 10 is a P pentomino. (One non-brute force way to see this is Pick's theorem.) So all the polygons are P pentominoes, and we're done.