# Why Area give the wrong results when the region cannot be formed?

Bug introduced in 11.1 or earlier and persisting through 11.2.0 or later

I use Area function to evaluate the area of the polygon. I found that when two vertex points are the same, the Area give the wrong result. For example, I define the following function:

areacoor[x_, y_] := Area@Polygon[{{x, y}, {0.5, 0.5}, {1, 1}}]


which is the area of the triangle whose vertex points is {x, y}, {0.5, 0.5} and {1, 1}. When $(x,y)$ move towards to any two other points, for example, $(x,y)\to (0.5,0.5)$, the area of the triangle will become smaller and smaller. When $(x,y)=(0.5,0.5)$, the triangle will become a line and the area should be 0. However areacoor[0.5,0.5] gives 0.707107. I wonder why I obtain this result, which I think is not reasonable.

• I think this is a bug. Maybe somebody tried to make the definition of Area rebust for degenerating polygons, deleted duplicate vertex coordinates and somewhat mixed up Area and RegionMeasure... Jan 26, 2018 at 9:10
• Added the bugs tag. Please report it to the tech support. Jan 26, 2018 at 11:32
• Area[Polygon[{{0.4, 0.4}, {0.5, 0.5}, {1, 1}}]], too, misfires. Jan 26, 2018 at 14:20
• in 11.1 other colinear point sets generate an (incomprehensible) error message. Only a "wrong" result for coincident points. Jan 26, 2018 at 16:03

areacoor[x_ , y_ ] :=Evaluate@Area@Polygon[{{x, y}, {0.5, 0.5}, {1, 1}}]

Plot3D [Area[Polygon[{{x, y}, {1/2, 1/2}, {1, 1}}] ], {x, 0, 1}, {y,0, 1}]