The authors consider the problem of when a finite sum of finite products of Toeplitz operators is a compact perturbation of a Toeplitz operator. The authors introduce a generalized area function associated with a finite sum of finite products of Toeplitz operators and establish a distribution function inequality for the area function. By using the distribution function inequality, they prove that a finite sum \(T\) of finite products of Toeplitz operators is a compact perturbation of a Toeplitz operator if and only if \(\lim_{| z| \to 1} \| T-T_{\phi_z}^\ast TT_{\phi_z}\| =0\), where \(\phi_z\) denotes the Möbius map \(\phi_z(w)=\frac{z-w}{1-\bar z w}\).