Please post the original image, not a screenshot. Is the actual image a real gray scale image, or is it that RGB computer graphics diagram? Then, how are you getting the black and white (binary, logical) image from the original image?
Anyway, once you have the image segmented so that you have a mask with only black pixels and white pixels, then you can use sum() and nnz() to do the count. Assume circleMask is the mask of the overall, outer circle, and mask is your segmented, binary image of stuff inside you're interested in, then
numPixelsInCircleMask = sum(circleMask(:))
numWhitePixelsInMask = nnz(mask)
numBlackPixelsInMask = nnz(~mask)
whiteAreaFraction = numWhitePixelsInMask / numWhitePixelsInMask
blackAreaFraction = 1 - whiteAreaFraction