One of the remarkable features of Beijing Daxing International Airport is the use of various curved surfaces. Describing the curvature of space is an important content of geometry research. It is stipulated that the curvature of the vertex of a polyhedron is equal to the difference with the sum of the face angles of the polyhedron at this point (the inner angle of the polyhedron is called the face angle of the polyhedron, and the angle adopts the arc system), and the curvature of the non-vertex on the polyhedron surface is zero. The total curvature of a polyhedron is equal to the sum of the curvatures of the vertices of the polyhedron. For example, each vertex of a regular tetrahedron has three face angles, and each face angle is, then the curvature of each vertex of a regular tetrahedron is, and its total curvature is.

(1) Find the total curvature of the four pyramids;

(2) If the polyhedron satisfies: number of vertices-number of edges+number of faces =2, it is proved that the total curvature of the polyhedron is constant.

Answer the question 1

A quadrangular pyramid has five vertices and five faces, including 1 quadrangles and four triangles, and the sum of their face angles is =

Total curvature =

Answer the second question.

As shown in the figure, an n-sided polygon on the plane can be divided into n triangles by taking any point Q in the polygon. Because the sum of the internal angles of any triangle is equal to 180 degrees, the sum of the internal angles of these triangles is equal to: because there is still an angle at point Q, the sum of the internal angles of the N-polygon is equal to:, that is:.

For each face of a polyhedron, we can get the number of triangles = the number of sides × 2.

Number of available angles = number of faces

According to the curvature formula,

Total curvature = number of vertices-(number of edges-number of faces) = (number of vertices-number of edges+number of faces)

The certificate is complete.

Regression teaching material

The sum of internal angles of polygons is a basic problem in geometry. The title of chapter 1 1 in Mathematics-Grade 8 (page 2 1) of People's Education Press is: the sum of interior angles of polygons.

It can be seen that this problem belongs to: basic concepts and basic methods.

Refine and improve

Why do many students find this problem difficult? The reason is that it is too basic. After a lot of repeated mechanical training, students can't solve problems with basic methods. When you encounter such a problem that doesn't depend on all the "problems", you have no way to start.

In order to successfully answer this question, candidates must pass several levels:

1) The key to understanding the topic is to understand a new concept: the curvature of a polyhedron is within a few minutes.

2) Mastering the derivation process of polygon internal angle sum formula, not just the conclusion.

3) After observation and induction, it is concluded that the number of triangles = the number of sides ×2. This is not difficult, but in reality, some people just can't do it.

Over the years, there has been a tendency that theory is divorced from practice in middle school mathematics teaching: experts constantly emphasize mathematical ideas and methods; The middle school teacher has been taking the students to brush the questions desperately.

The entrance examination of mathematics in eight provinces sends a signal to everyone: there are many ways to propose marriage. Under the premise of not changing the college entrance examination system, it is entirely possible to strengthen the examination of students' ability.

For students and teachers preparing for exams, my suggestions are:

1) Think more and summarize more; Never do the problem blindly.

2) Spend some time reading textbooks, including junior and senior high school textbooks, which will be used.