Linear spectral unmixing is a widely used technique in hyperspectral remote sensing to quantify materials present in an image pixel. In order to produce accurate estimates of abundances, nonnegativity constraint and sum-to-one constraint must be imposed on the abundances of materials. Under these two constraints, linear spectral unmixing is often formulated as a convex optimization problem that requires more advanced optimization technology, leading to excessive computational complexity. In this paper, a novel geometric method is presented for solving the fully constrained linear spectral unmixing problem. Specifically, abundances are first expressed as the ratios of signed volumes of simplexes. Then, Laplace expansion is applied in the process of determinant calculation, which derives a new low-complexity abundance estimation method. Furthermore, the mixed pixel outside the simplex is iteratively projected onto the facet planes through the endmember vertices for making the abundances satisfy the nonnegativity constraint. This process is continued until one finds a projected point lying inside the simplex. The proposed method is in line with the least squares criterion. Experimental results based on simulated and the AVIRIS Cuprite data sets demonstrate the superiority of the proposed algorithm with respect to other state-of-the-art approaches.