Finding area of intersection regions

Question

I want to be able to find the area of the intersection between a hexagon and a slice. I have tried RegionMeasure or Area but I seem to get incorrect answers. RegionMeasure gives length when the intersection is a line, so I switched to Area but sometimes it still gives weird answers, when the intersection happens to be a line.

One reason why I think this might occur is that I build the hexagon using Polygon, but the "slice" is with Disk. Is that the problem? How to I make a region as a slice, so that Area[RegionIntersection[hexagon, slice]] between the two regions will always give $0$ for a line intersection and no intersection, i.e. return a nonzero result only for 2D intersection.

See picture. I want to know the area of intersection of the black hexagon will each of the 6 (colored for visualization) slices below. The answer should be non-zero for the 2 upper left slices, and 0 for every other slice, even though some have "line intersections".

Thanks.

EDIT: The problem is that edges get non-zero measure. Why is the below code giving .7, and not 0?

DiskR = 1.5;
Show[
  Graphics[{Orange, Opacity[0.35], 
  Disk[{0, 0}, DiskR, {Pi/6, 3 Pi/6}]}], 
  Graphics[Polygon[{{0, 0}, {0, .7}, {-1, 1}}]]]

and

Area[
 RegionIntersection[
  Polygon[{{0, 0}, {0, .7}, {-1, 1}}], 
  Disk[{0, 0}, DiskR, {Pi/6, 3 Pi/6}]
 ]
]

Answer 1

In this case, the intersection between two 2D regions is a 1D region - a line. It makes sense that the functions that are called by RegionMeasure and RegionIntersection might be influenced by roundoff errors associated with using floating point numbers rather than exact numbers.

So a workaround is to convert your expressions to use exact numbers prior to taking the area. I had originally used Rationalize to do this, and it works in this case, but it doesn't always. So we can borrow a function from illian that is a bit more robust

sp = Function[p, SetPrecision[p, Infinity]];    
DiskR = 1.0; 
Area[
 RegionIntersection[sp@Polygon[{{0, 0}, {0, .7}, {-1, 1}}], 
  sp@Disk[{0, 0}, DiskR, {Pi/6, 3 Pi/6}]]]

(* 0 *)

While this behavior is understandable, I'd still say it's a bug