The study of solids is a natural science of spatial figures, and solids are the most suitable objects to apply scientific and analytic methods in school mathematics. On the other hand, the study of solids has several difficulties. In this paper we introduce the concept of acuteness of a solid, and a method measuring acuteness of solids. We expect that they bring us lessons of abundance in school mathematics.
Let X be a convex polyhedron, and A a vertex of X. Suppose that k faces meet at the vertex A and whose interior angles at A are θ1, θ2, …, θk. Then we define the the deficit of the angle c(A) at the vertex A by 360° − (θ1 + θ2 + … + θk). By the hypothesis of convexity, we have c(A) ≧ 0. This c(A) measures the acuteness of the polygon X at the vertex A, and has several nice properties. For example, by using the Euler formula for convex polyhedra, we see that the sum of the deficits of a convex polyhedron is equal to 720°.