On page 345 of Astronomical Algorithms by Meeus, he states that the illuminated fraction $k$ of the Moon's disk is given by the ratio $BC:AC$ (see diagram) where $BC$ is the width of the illuminated crescent and $AC$ is the Moon's diameter. He then goes on to say that this is the same ratio as the crescent's area ($NCSB$) to the total disk area ($NCSA$). The illuminated fraction is given by
$$k=\frac{1+\cos i}{2},$$where $i$ is the phase angle, the angle between the Sun and Earth as seen from the centre of the Moon. Now, I can understand (and derive) why $k$ is the ratio of the lengths $BC:AC$, but I cannot see why it is also the ratio of the areas. I've looked at the (complicated, to my eyes) formula for the area of a lune (a crescent shape), given in Wikipedia and can't make the link between it and the total disk area. Thanks.
Hint:
$NBS$ is not a part of circle but a half on an ellipse with semiaxes $ON$ and $OB$. The rest is trivial knowing the expression for the ellipse area