Precise quantification of complex three-dimensional (3D) structures from laser scanning microscopy (LSM) images is increasingly necessary in understanding normal function and pathologic processes in biology. This protocol describes a versatile shape analysis algorithm, Rayburst sampling, that generates automated 3D measurements from LSM images. Rayburst defines and efficiently casts a multidirectional core of rays from an interior point to the surface of a solid, allowing precise quantification of anisotropic and irregularly shaped 3D structures. Quantization error owing to the finite voxel representation in digital images is minimized by interpolating intensity values continuously between voxels. The Rayburst algorithm provides a primitive for the development of higher level algorithms that solve specific shape analysis problems. Examples are provided of applications to 3D neuronal morphometry: (i) estimation of diameters in tubular neuronal dendritic branching structures, and (ii) measurement of volumes and surface areas for dendritic spines and spatially complex histopathologic structures.