I don't know of any way to calculate this stuff for an irregular shaped polygon, but it might be possible... However, if it is impossible, how's about trying to divide it into squares and triangles? Draw a scale map, use one wall as base line, move lines from there, with 90 degree angles...
Or it may be easier to surround the shape with a square or box, and subtracting the area outside.
Oh, you do need the angles of the walls for this.
(hmm... in theory, you could flood the entire room with water, measure how many gallons you put in, and compensate for the height of the room...
)