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We present a quantitative method for determining the viscoelastic properties of materials with nanometer spatial resolution. The approach is based on the atomic force acoustic microscopy technique that involves the resonant frequencies of the atomic force microscopy cantilever when its tip is in contact with a sample surface. We derive expressions for the viscoelastic properties of the sample in terms of the cantilever frequency response and damping loss. We demonstrate the approach by obtaining experimental values for the storage and loss moduli of a poly(methyl methacrylate) film using a polystyrene sample as a reference material. Experimental techniques and system calibration methods to perform material property measurements are also presented.