[zakthor]

Well... I thought about this and tried to come up with a cool solution using infinite series but I failed so here's a regular way and I can show my work.

The height from the middle of the line to the circle is H.
The distance between the two points (P0, P1) on the arc is D.

D/2 is the length to the middle of the span, I'll call that point D2.

The distance from that point in the middle of span up to the circle is H and I'll call that point H2.

The distance from H2 to the center of the circle is R - the radius of the circle.

Therefore the distance from D2 to the middle of the circle is... R - H.

So now we have a right triangle and two lengths:
P0 to D2 is length D/2.
P0 to middle of circle is R.
D2 to middle of cicle is R-H.

By pythagoras: R^2 = (D/2)^2+(R-H)^2

Expand to:

R^2 = D^2/4 + R^2 - 2RH + H^2

Cancel the R^2

0 = (D^2)/4 - 2RH + H^2

2RH = (D^2)/4 + H^2

R = (D^2)/8H + H/2

We have the circle's radius, now we need to know the angle between P0, P1 which is a trig identity:

Theta = 2 asin(D/2R)

Theta = 2 asin(D/((D^2)/8H + H/2))

Arc length is: R Theta

Plugging it all in you get:

S = ((D^2/4H) + H) arcsin(D / ((D^2 / 4H) + H))

S = 80.25^2/(4 x 2.125)+2.125 x arcsin(80.25/((80.25^2 / (4 x 2.125))+2.125

S = 80.45"

The arc is 0.2" longer than the distance of the span.

I'm still thinking about a different way to do this but probably its a dead end. Anyway there you go!