We study the distribution of domain areas, areas enclosed by domain boundaries ("hulls"), and perimeters for curvature-driven two-dimensional coarsening, employing a combination of exact analysis and numerical studies, for various initial conditions. We show that the number of hulls per unit area, n_h(A,t) dA, with enclosed area in the interval (A,A+dA), is described, for a disordered initial condition, by the scaling function n_h(A,t) = 2c_h/(A + λ_h t)^2, where c_h=1/8π√(3)≈ 0.023 is a

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... al constant and λ_h is a material parameter. For a critical initial condition, the same form is obtained, with the same λ_h but with c_h replaced by c_h/2. For the distribution of domain areas, we argue that the corresponding scaling function has, for random initial conditions, the form n_d(A,t) = 2c_d (λ_d t)^τ'-2/(A + λ_d t)^τ', where c_d=c_h + O(c_h^2), λ_d=λ_h + O(c_h), and τ' = 187/91 ≈ 2.055. For critical initial conditions, one replaces c_d by c_d/2 (possibly with corrections of O(c_h^2)) and the exponent is τ = 379/187 ≈ 2.027. These results are extended to describe the number density of the length of hulls and domain walls surrounding connected clusters of aligned spins. These predictions are supported by extensive numerical simulations. We also study numerically the geometric properties of the boundaries and areas.