解题方法
1 . 如图,在多面体
中,平面
平面
,
平面
,
和
均为正三角形,
,
.
(1)求多面体
的体积.
(2)在线段
上是否存在点
,使得
平面
?说明理由;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9142a8490de14a87eda628ffa7e28982.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17580410bf63dba4fe164265afaac4cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/662698361c6b3ddaf0c28a3c87be53e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ac451db3443cabb204f96c31fd4a02e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8a7b5adfcac0f46a4cd19da4ebb4a2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75929268210da5976bc37d080da030dd.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/8/7/9ffb5c30-d01b-49fa-a02b-ecebba9e6db3.png?resizew=141)
(1)求多面体
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9142a8490de14a87eda628ffa7e28982.png)
(2)在线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c8ccd4181f956f6e0140bf0ab8f0716.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9a32bd7a1b78b5a0ec562c4025aea8c.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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3 . 在
平面上,将两个函数
和
、两条直线
和
围成的封闭图形记为
,如图所示,记
绕
轴旋转一周而成的几何体为
,则
的体积值________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78b91d90e35deba1cdc76b3247d8d909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/034e4d08e558565864c14672bf75a4ec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/107babba45f110012183dc4dc54490f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2a7df955fc17e92fd86302f8c34664a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cffa35373ec4e4684107b42adb7a5161.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cffa35373ec4e4684107b42adb7a5161.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/11/81df9384-c6f4-4fd4-bc9c-cf0544733343.png?resizew=205)
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解题方法
4 . 已知2023年第57届世界乒乓球锦标赛规定适用的乒乓球直径为4cm.如图,是一个正方形硬纸板,现有同学将阴影部分裁掉,把剩余的扇形部分制作成一个圆锥型的纸筒.若这样的乒乓球能够完全装入该同学所制作的圆锥型的纸筒内,则正方形纸板面积的最小值为________ 平方厘米.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/11/06fd5095-890c-4300-a7cd-403f2f85be49.png?resizew=135)
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2023-07-09更新
|
334次组卷
|
2卷引用:上海市复旦大学附属中学2022-2023学年高一下学期期末数学试题