Area of an ASCII polygon
Pyth, 47 46 45 36 30
FNs.zx=Z}N&quot;\/L&quot;aY|}N&quot;\/V&quot;yZ;sY

Explanation:
FNs.z            For every character in input, except newlines...
  x=Z}N&quot;\/L&quot;     Swap state if /, \, or L.
  aY|}N&quot;\/V&quot;yZ;  Append 1 if /, \, or V, else 2 times the state to Y.
sY               Sum Y and print.

We have two states, &quot;in the polygon&quot;, and &quot;out of the polygon&quot;. The following characters each do the following when reading them from top-left to bottom-right:
/ \     swap state, add one to area
V                   add one to area
_ space             if in polygon, add two to area
L       swap state, if in polygon, add two to area

Note that &quot;add one to area&quot; and &quot;if in polygon, add two to area&quot; are mutually exclusive.