From the Youtube, I noticed that the Klein bottle can be sliced into two Mobius strips. Or this experiment is also explained in this following link: http://www.kleinbottle.com/sliced_klein_bottles.htm
So, given that I got the surface area of the Mobius strip, can I calculate the surface area of the Klein bottle? But the problem here, I think is, that the Klein bottle used in this experiment actually is the 3 dimensional one with the self-intersecting part, while the theoretical and ideal Klein bottle only lives in 4 dimensional space without self-intersection.