[I adapted the question to reflect what I've learned from Alexandre's answer: that Euclid never talks of lengths and areas but only of line segments and figures (like squares). The question itself remains unchanged.]
For Euclid, magnitudes were things that can be compared, be equal, smaller or greater than each other, that can be added and subtracted, and that can have proportions. (Next to these, there were numbers, another type of thing with the same properties.)
There are four magnitudes Euclid explicitly and mainly deals with:
lengths ofstraight line segments
angles between straight line segments
areas ofplane figures (polygons and circles) volumes ofsolid figures (polyhedra and spheres)
lengths of straight line segments can be multiplied (giving areas and volumes figures), as well as numbers (giving numbers).
He also deals with
lengths of circle segments, at least with the length of the whole circle (circumference).
But does he anywhere talk explicitly about
the length ofarbitrary circle segments and compares, adds, or substracts them?
Furthermore: Was he aware that an angle is "proportional" to
the length of a corresponding circle segment and did make use of this (which would not have meant for him, that the two were the same)?