Hausdorff measure is often used to measure fractal sets. However, there is a more natural quantity, Minkowski content, which more closely matches the scaling limit of discrete counting measures and is closely related to the idea of local time. I will discuss this in the context of several sets for which Chris Burdzy made fundamental contributions: cut points for Brownian paths and outer boundary of two-dimensional Brownian motion. The latter is closely related to the Schramm-Loewner evolution (SLE). I will include joint work with Mohammad Rezaei and recent work with Nina Holden, Xinyi Li, and Xin Sun.