Foreword
Typical Example Analysis
Example 1
[2019 college entrance examination paper science of mathematics new curriculum standards Ⅱ Question 16] China has a long cultural stone, stone seals is one of the representatives of culture. The shape of a rectangular parallelepiped more seals, cube or cylinder, but the shape of the letter Tokgo official seals during the Northern and Southern "semi-regular polyhedron", semi-regular polyhedron is surrounded by two or more polyhedral regular or into, Semipositone polyhedron reflecting symmetry of a mathematical, FIG. 2 is a rib number \ (48 \) semi-regular polyhedron, its all vertices on the same surface of a cube, and this cube edge length \ (1 \) , the total __________ semi-regular polyhedron faces, an edge line length of __________.
Analysis: process of making a semi-regular polyhedron, as shown below;
Analysis: If we regard it as a three-layer, each layer has the \ (8 \) faces, plus another two upper and lower faces, so that a total of \ (3 \ times 8 + 2 = 26 \) faces .
As shown, the edge length is set \ (X \) , i.e. \ (the MN = NE = X \) , the \ (\ triangle EHN \) isosceles right triangle,
By the \ (NE = X \) , then the clear \ (= NH2 \ cfrac {\ sqrt {2} {2}} X \) , and \ (+ 2NH the MN. 1 = \) ,
The \ (X + 2 \ Times \ cfrac {\ sqrt {2}} = {2}. 1 X \) , i.e. \ ((\ sqrt {2} + 1'd). 1 X = \) , solve for \ (x = \ sqrt {2} -1 \) .
综上可知，此半正多面体共有\(26\)个面，棱长为\(\sqrt{2}-1\)。
【解后反思】
1、求其表面积；
2、求其体积；
3、求其内切球的半径；
4、求其外接球的半径；