Explanation of Numerical Aperture

I am being introduced to fibre optics in the class and all text books I feffer to derrive Numerical Aperture by applying pythogaras formula on a triangle having refractive indices as it's sides.

NA=\sqrt{n12+n22}

How can the refractive indices be proved to be proportional to the dimentions of the triangle...

Moreover, does the numerical aperture depend on the dimentions of the cable?

hi
the numerical aperture indicates the ability of the fiber of the amount of power it can take from the src and sending to the rx by guided it only in the core.
as we know we have different types of fibers and every type has its properties

for example
step index fiber >> the numerical aperture doesn't depend on the dimentions of the fiber

graded index fiber>>the numerical aperture depend on the dimentions of the fiber

NOteNA=sqrt{n12-n22} not NA=sqrt{n12+n22}

Added after 1 minutes:

hi
the numerical aperture indicates the ability of the fiber of the amount of power it can take from the src and sending to the rx by guided it only in the core.
as we know we have different types of fibers and every type has its properties

for example
step index fiber >> the numerical aperture doesn't depend on the dimentions of the fiber

graded index fiber>>the numerical aperture depend on the dimentions of the fiber

NOteNA=sqrt{n12-n22} not NA=sqrt{n12+n22}