For example, let's look at the following ASCII art:
/\ - encloses a total of 2 cells - half a cell per slash \/ /\ - encloses 4 cells fully and half of 8 cells, for a total of 8 / \ \ / \/
Your challenge is to write a program that determines (and outputs) the total area enclosed by ASCII art composed only of spaces, slashes and newlines. Slashes won't necessarily be a part of a shape with non-zero area. A point is defined as enclosed iff it is unreachable from any point outside the art's bounding box, if slashes are impassable.
Slashes have zero area and cells are assumed to be \$1\times1\$ squares.
/s represent lines that connect the lower left corner with the upper right corner, and
\s represent lines the connect the upper left corner with the lower right corner of the corresponding cells. Spaces represent empty space.
/\/\ \ \ \/\/
Encloses 3 cells fully and 10 partially, for a total of 8.
///\\\ // \\ / /\ \ \ \/ / \\ // \\\///
Encloses 12 cells fully (the four innermost slashes have both corresponding half-cells enclosed) and half of 12 cells, for a total of 18.
/\/\ /\/\ /\/\
Encloses 0 cells.
/\ / \ \/\/ /\/\ \ / \/
Encloses 8 cells fully and 12 cells partially, for a total of 14.
/\ / \ / \ \ / / \ \ \/ \/ \ \ / \ \ \/\/
Encloses 25 cells fully and half of 18 cells, for an area of 34.
This is tagged code-golf, so the shortest answer wins.