A well known Illustrator trick (which has been shown also here in GDSE several times) to separate the wanted length from a path which has a known total length is to apply originally straight line as Art Brush. The straight line is as long but divided to 2 separate differently colored parts.
Lets assume curve AB is 5 inches long. The green line is 2 inches long and the red line is 3 inches long. The colored lines are dragged together to the brushes collection and defined to be an Art Brush which should be stretched to full length.
The brush is applied to AB, Object > Expand Appearance is applied and the result is Ungrouped twice. The green part of AB is now separate and it's 2 inches long.
Another way to apply the same base idea is to divide AB to known proportions. If the lines were both as long - no matter how long exactly, but as long, AB whould be divided to 2 as long parts.
There's harmfully extra nodes appeared to the green part - probably to make the math easier in Illustrator. If that cannot be accepted you can place the original AB on the top of the green part; the left end nodes should snap perfectly if you have smart guides and snap to point ON and drag the selected curve from the left end with the direct selection tool:
Be sure the original is on the top. Apply the Scissors. They snap perfectly if you have smart guides and snap to point ON. The separated part of the original is free immediately and it has got no new intermediate nodes:
BTW. As others have already suggested you should check if the glass frame is constructed from straight lines and circular arcs in a simple way - there's a half circle which crosses straight radial lines. In that case you can use angles to find the arc lenghts. That's nothing special, only a rewrite of the fact that plane angle measure is originally in elementary geometry defined with arc length.
The length af the green arc from the midline of ray A to the midline of ray B is
(alpha/180 degrees) * Pi * R
where alpha = the angle between rays A and B. That's the same part of the half circle length Pi * R as what proportion alpha is of 180 degrees.
To get how long piece of the arc is between the red rays you subtract the thickness of the rays.