#Pyth, 47 46 45 36 30

FNs.zx=Z}N"\/L"aY|}N"\/V"yZ;sY

Explanation:

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

We have two states, "in the polygon", and "out of the polygon". 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 "add one to area" and "if in polygon, add two to area" are mutually exclusive.

Pyth, 47 46 45 36 30

FNs.zx=Z}N"\/L"aY|}N"\/V"yZ;sY

Explanation:

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

We have two states, "in the polygon", and "out of the polygon". 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 "add one to area" and "if in polygon, add two to area" are mutually exclusive.

#Pyth, 47 46 45 36 30

FNs.zx=Z}N"\/L"aY|}N"\/V"yZ;sY

We have two states, "in the polygon", and "out of the polygon". The following characters each do the following when reading them from top-left to bottom-right:

\n  nothing
/ \     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 if you split on \n and sum the lines, \n disappears, and the "add one to area" and "if in polygon, add two to area" become mutually exclusive.

#Pyth, 47 46 45 36 30

FNs.zx=Z}N"\/L"aY|}N"\/V"yZ;sY

Explanation:

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

We have two states, "in the polygon", and "out of the polygon". 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 "add one to area" and "if in polygon, add two to area" are mutually exclusive.

#Pyth, 47 46 45

M?1}H"\/V"*yG}H"L _"aY0u+GgeaYxeY}H"\/L"Hs.z0

We have two states, "in the polygon", and "out of the polygon". The following characters each do the following when reading them from top-left to bottom-right:

\n  nothing
/ \     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

#Pyth, 47 46 45 36 30

FNs.zx=Z}N"\/L"aY|}N"\/V"yZ;sY

We have two states, "in the polygon", and "out of the polygon". The following characters each do the following when reading them from top-left to bottom-right:

\n  nothing
/ \     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 if you split on \n and sum the lines, \n disappears, and the "add one to area" and "if in polygon, add two to area" become mutually exclusive.