In this work, we study the coercivity of a family of quadratic finite volume element (FVE) schemes over triangular meshes for solving elliptic boundary value problems. The analysis is based on the standard mapping from the trial function space to the test function space so that the coercivity result can be naturally incorporated with most existing theoretical results such as $H^1$ and $L^2$ error estimates. The novelty of this paper is that, each element stiffness matrix of the quadratic FVE schemes can be decomposed into three parts: the first part is the element stiffness matrix of the standard quadratic finite element method (FEM), the second part is the difference between the FVE and FEM on the element boundary, while the third part can be expressed as the tensor product of two vectors. As a result, we reach a sufficient condition to guarantee the existence, uniqueness and coercivity result of the FVE solution on general triangular meshes. Moreover, based on this sufficient condition, some minimum angle conditions with simple, analytic and computable expressions are obtained. By comparison, the existing minimum angle conditions were obtained numerically from a computer program. Theoretical findings are conformed with the numerical results.