1. Common stereo graphics
(1) cylinder
(1) prism: two faces are parallel to each other, the other face is a quadrilateral, and the public sides of every two adjacent quadrilaterals are parallel to each other. The geometry enclosed by these surfaces is called prism, such as triangular prism, quadrangular prism, pentagonal prism, etc.
(2) Cylinder: The geometric body formed by taking the straight line where one side of a rectangle is located as the rotation axis and other sides rotating around it is called a cylinder.
(2) Cone
① Pyramid: One face is a polygon and the other faces are triangles with a common vertex. The geometric shapes enclosed by these faces are called pyramids, such as triangular pyramids, quadrangular pyramids and pentagonal pyramids.
Cone: the geometric shape formed by taking the straight line where one side of a right triangle is located as the rotation axis and the other sides rotating around it is called a cone.
(3) Sphere: A semicircle takes its diameter as its axis of rotation, and the rotated geometry is called a sphere.
Example 1 Judge whether the following statement is correct:
(1) The sizes of the upper and lower surfaces of the cylinder are different ().
(2) The bottoms of cylinders and cones are all circles ().
(3) The bottom surface of the prism is not necessarily a quadrilateral ().
(4) The side of the cylinder is a plane ().
(5) The sides of the pyramid are not necessarily triangles ().
Analysis: The top and bottom surfaces of the cylinder are parallel and equal (same shape and equal size), so (1) is wrong; The top and bottom surfaces of the cylinder are round, and the bottom surface of the cone is round, so (2) is correct; Prism can be triangular prism, quadrangular prism, pentagonal prism, etc. That is, the bottom surface of a prism is not necessarily a quadrilateral, so (3) is correct; The side of the cylinder is curved, not flat, so (4) is wrong; The sides of the pyramid must be triangles, so (5) is wrong.
Answer: (1)× (2)√ (3)√ (4)× (5)
2. Classification of 3D graphics
Stereographic figure
In order to facilitate understanding and memory, the classification of three-dimensional graphics is summarized as follows:
Example 2 The number of columns in the figure below is ().
A. 1B.2C.3D.4
Analysis: Cylinders are characterized by parallel upper and lower bottoms (same shape and size), so ① and ② are judged as cylinders.
Answer: b
3. Polyhedron
(1) The concept of polyhedron: The surface surrounded by prisms and pyramids is a plane, and a three-dimensional figure like this is called a polyhedron.
As shown in the figure, the following figures are: prism (cuboid) and pyramid (triangular pyramid), both polyhedrons.
(2) Regular Tetrahedron: A spatial figure surrounded by four identical regular triangles is called a regular tetrahedron, and the vertices and edges of these triangles are called the vertices and edges of a regular tetrahedron respectively (the common edge of adjacent triangles is only one edge).
(3) Regular hexahedron: Similarly, the vertices and edges of each square forming a cube are called the vertices and edges of a regular hexahedron respectively (the common edges of adjacent squares are only counted as one edge).
There are also octahedron, dodecahedron and icosahedron, as shown in the figure.
Common polyhedral prisms and pyramids are polyhedrons, while cylinders, cones and spheres are not polyhedrons.
The bottom of the prism is pentagonal. How many sides and vertices does it have? * * * How many faces are there?
Analysis: It is easy to know that a three-dimensional figure is a pentagonal prism. Just answer the questions with pictures.
Solution: It has 5 sides, 10 vertices, and * * * has 7 faces.
Determination of the number of edges, vertices and faces of an analytic regular prism A prism with a bottom surface of n faces has n sides, 2n vertices and (n+2) faces.
4. The characteristics of general geometry
geometry
the seamy side
One side, one side
Vertex number
column
The two bottom surfaces are parallel and equal in shape and size.
curved surface
not have
circular cone
The bottom surface is round.
curved surface
one
prism
Two polygons with parallel bottoms and equal shapes and sizes.
plane
have
pyramid
The bottom surface is polygonal.
plane
have
The triangular prism has five faces, six vertices and nine sides. The number of faces of a quadrangular prism is 6, the number of vertices is 8, and the number of edges is 12. Similarly, the number of faces of an N prism is n+2, the number of vertices is 2n, and the number of edges is 3n.
The triangular pyramid has four faces, four vertices and six sides. A pyramid has five faces, five vertices and eight edges. Similarly, the number of faces of an N-pyramid is n+ 1, the number of vertices is n+ 1 and the number of edges is 2n.
Example 4 How many faces are surrounded by two geometric bodies in the figure? How many lines intersect? Are they straight or curved?
( 1) (2)
Analysis: Look at the geometry in this problem carefully. (1) is made of a cylinder cut longitudinally along its high line. Because the side of a cylinder is a surface, the side of this geometry is also a surface. (2) Cut an angle to form a hexahedron, and all the surfaces that make up the geometry are planes.
Solution: Geometry (1) in the graph is surrounded by four faces; Faces intersect into six lines, four of which are straight and two are curved.
The geometric body (2) is surrounded by seven faces; Faces intersect into 14 lines, all of which are straight.
5. Euler formula
By calculating the number of vertices (V), faces (F) and edges (E) of a regular polygon, we can draw a conclusion:
polyhedron
V
F
E
regular tetrahedron
four
four
six
regular hexahedron
eight
six
12
regular octahedron
six
eight
12
pyritohedron
20
12
30
Regular icosahedron
12
20
30
As can be seen from the above table, the relationship among the number of vertices, faces and edges of a polyhedron is v+f-e = 2, that is, the number of vertices+faces-edges = 2. The great mathematician Euler proved this formula, so people call it Euler formula.
When using the formula "V+F-E = 2", we should correctly judge two of the vertices, faces and edges. The number of faces of a polyhedron is known, which is consistent with the name of the polyhedron. For example, in the above table, the number of faces of tetrahedron is 4, the number of faces of octahedron is 8 and the number of faces of dodecahedron is 12. Therefore, only one of the vertices and edges needs to be known.
When a cube is cut off, the rest is still a polyhedron. Although the number of vertices, edges and faces will change, the relationship between them remains unchanged and still conforms to Euler formula.
The application of Euler formula in solving the quantitative relationship among edges, vertices and faces of polyhedron is relatively simple and convenient. In order to get the quantitative relationship between vertices, edges and faces of polyhedron, we can analyze the data in the table in detail.
Example 5 is a cube as shown in figure 1. Cut a piece with possible patterns ②, ③, ④ and ⑤.
(1) We know that the cube in Figure ① has 8 vertices, 12 edges and 6 faces. Please fill in the number of vertices, edges and faces of this block in Figures ②, ③, ④ and ⑤ in the table below.
draw
Vertex number
Edge number
Number of faces