Here length means 1-Hausdorff measure. This seems to be known, what is the reference? Or very short proof?



Connected planar compact set with finite length is path connected
See Exercise 3.5 in "The geometry of fractal sets" by K. J. Falconer. 


It says that any such set is an image of rectifiable curve.


For a short proof, check "Rectifiable curve" in [my collection](https://arxiv.org/abs/0906.0290).


P.S. The earliest reference I found: Theorem 2 in *Continua of finite linear measure. I.* by Eilenberg and Harrold (1943).


