*also affiliated with Kavli Institute for Theoretical Physics, University of California, Santa Barbara Graphene provides an ideal system to test the statistical mechanics of thermally fluctuating elastic membranes. Its high Young’s modulus means that thermal fluctuations are already important at nanometer length scales, dramatically modifying the mechanical properties of thermalized graphene sheets. In particular flexural phonons - escape into the third dimension - lead to a strongly length-scale dependent stiffening of the microscopic bending rigidity. We study the effect of thermal fluctuations on graphene ribbons of width W and length L, pinned at one end, via coarse-grained Molecular Dynamics simulations and renormalization group theory and compare with analytic predictions of the scaling of width-averaged root-mean-squared height fluctuations as a function of distance along the ribbon. Scaling collapse as a function of W and L also allows us to extract the scaling exponent governing the long-wavelength stiffening of the bending rigidity. A full understanding of the geometry-dependent mechanical properties of graphene, including arrays of cuts and holes (graphene kirigami), may pave the way for the design of a variety of pure graphene modular elements with prescribed mechanical properties.