解题方法
1 . 将一个边长为2的正六边形
(图1)沿
对折,形成如图2所示的五面体,其中,底面
是正方形.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/30/789c90f8-a658-4d0b-a39f-4b279dfeabea.png?resizew=439)
(1)求二面角
的大小.
(2)如图3,点
分别为棱
上的动点.求
周长的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9165d9bfbb0f0d19eb482c2a4c1b29b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad3a079cfdcca9acdacecbf08f9f78cc.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/30/789c90f8-a658-4d0b-a39f-4b279dfeabea.png?resizew=439)
(1)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2def0d393ca995cfe6e2deb25fb35d3.png)
(2)如图3,点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c0f067a2a348ceb24a408f82992eab8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bb025beb66dc609261deac78327954c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59f3f9f4edf520ce61c8e83a2be394d6.png)
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2 . 如果一个正多面体的所有面都是全等的正三角形或正多边形,每个顶点聚集的棱的条数都相等,这个多面体叫做正多面体.有趣的是只有正四面体、正方体、正八面体、正十二面体和正二十面体五种正多面体,现将它们的体积依次记为,
.
(1)利用金属板分别制作正多面体模型各一个,假设制作每个模型的外壳用料(即表面积)均等于
,分别求出
和
的值;并猜想
与
的大小关系(猜想不需证明)
(2)多面体的欧拉定理:简单多面体的面数
、棱数
与顶点数
满足:
.已知正多面体都是简单多面体,设某个正多面体每个顶点聚集的棱的条数为
,每个面的边数为
,求
满足的关系式;并尝试据此说明正多面体仅有五种.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5142a5f4db2068493b7d414806f24e5.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/11/b1197459-0825-43ff-858f-f8721faa0bc7.png?resizew=548)
(1)利用金属板分别制作正多面体模型各一个,假设制作每个模型的外壳用料(即表面积)均等于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bf956f7cef485a7a509fd8229d7eb48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/789ce79353afd7894c4a912815e370f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0af6f28b405604706431065a6620423.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fee8e86607a073a323a51640d0e40532.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1104314b67a6607d116064c8dd1a0108.png)
(2)多面体的欧拉定理:简单多面体的面数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be54e84508decfcce6d2fcbe6c8c1a92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a098e3851f80b3d3c273d34416c4778e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2442dc47b9650e00a0cef190e4cc5e5f.png)
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