If all of the sprockets are the same size....
Label each sprocket in your diagram A, B, and C (doesn't matter the order)
Distance from the center (axle) of A to the center (axle) of B = X
Distance from the center (axle) of B to the center (axle) of C = Y
Distance from the center (axle) of C to the center (axle) of A = Z
- These are obvious lengths
Circumference of a single sprocket = W
- This length is not so obvious initially, but think about looping a belt around only two sprockets. The only length of belt where length is not obvious is around the circumference of the sprockets. When the belt is wrapped around two sprockets, half of the circumference of each sprocket is where the additional belt length is. So this additional total length is one circumference of a sprocket.
-Move this into a three sprocket design such as the climber. Provided all of the sprockets exist on the inside of the loop and are of identical size, no additional surface area of a sprocket could be in contact with any additional belt/track surface of value. Every bit of the surface that is touched by the third sprocket, due to the angle between sprockets, is removed from the total length of belt wrapped around the first two, proportional to the angle of each.
X + Y + Z + W = length of your required belt
I would over estimate length by a small percentage and add a belt tensioner to the design for ease of installation and removal (and a measurement fudge factor).
If the sprockets are of different sizes, 'W' becomes MUCH harder to calculate. It would require ratios of circumferences to angle measurements. This makes my head hurt and I would STRONGLY suggest you over estimate and add a belt tensioner.
Disclaimer: That is the closest I could figure. I am neither a geometry guy nor an experienced tracked robotics guy. This just seemed the most logical to me.