The sum of equally spaced chords?

Let n denote the number of lines you are using.

If n is small, I do not think you can do better than what you are doing: a messy-looking sum of lengths, one for each line segment in the sum. A nice, easily-worked-with-by-hand closed form is probably not possible. But a calculator or computer would have no trouble evaluating an exact formula for small n.

For large n a useful approximation comes to mind. If you have taken a circle of radius r, and overlaid n equally spaced parallel lines over it from one end to the other, then the distance between any two consecutive lines is [the diameter of the circle]/n = (2r)/n. So if you multiply the sum of the lengths of the lines by (2r)/n you get a quantity with geometric meaning.

((2r)/n) * [the sum of the lengths of the line segments]

= the sum of ((2r)/n) * [the length of each line segment]

= the sum of [the area of a rectangle whose width is (2r)/n and whose height is the length of a line segment]

which is the area of a sort of "rectangular approximation" to the circle. If n is large in comparison to r, this rectangular region will be very close to the circle, so its area will be approximately the same as the area of the circle. Writing ~= for "approximately equal to", you can therefore expect ((2r)/n) [the sum of the lengths] ~= pi r^2, and hence

[the sum of the lengths] ~= (n pi r)/2

when n is large in comparison to r.

Not knowing anything about the specifics of the application, I'd recommend using the exact formulas (and a computer) if n is small in comparison to r, and otherwise, using this approximation. Even when n/r is only about as big as 10 or so, the approximation is already OK; if n/r is like 100, it is quite good.