When you get a binary image, You just add all the pixels in the image. Then you get the no of pixels with white. Subtract it with the total no of pixels (length x breadth), you get the no of black. You can take percentage, ratio or whatever you want now.
As for the area of the white pixels of the image, considering you have just counted the number of pixels, the unit will be (1)pixels (for a simple analysis) (2) if you want in SI units, then you have to multiply with the image sensor resolution which will give you the area in square meter.
Hope it helps.