2023高二上·上海·专题练习
解题方法
1 . 已知一个表面积为
的正方体的四个顶点在半球的球面上,四个顶点在半球的底面上,求半球的表面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eab3c56d6b15852879446ec184863d40.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/6/2e8e860f-d6c0-408d-80d2-a68c2c650ca3.png?resizew=144)
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2 . 已知圆锥的顶点为
,
为底面圆心,
,异面直线
与
所成角的余弦值为
,
的面积为
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/12/6282a5fe-9f1b-4ba2-9b49-9c65dbb578de.png?resizew=147)
(1)求该圆锥的表面积;
(2)求该圆锥内半径最大的球的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/950e673dc286ecdfa54dc7ab146770b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a6e2867f32d3f1c3cd36cd3a11a8580.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e6486784415f3537c9a13556c05d893.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72e49817548cb45b3d1e58570644c6fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/461dc722a8b3edc8147bf7b5f6e6eb11.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/12/6282a5fe-9f1b-4ba2-9b49-9c65dbb578de.png?resizew=147)
(1)求该圆锥的表面积;
(2)求该圆锥内半径最大的球的体积.
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2023-12-12更新
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222次组卷
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2卷引用:浙江省强基联盟2023-2024学年高二上学期12月联考数学试卷
解题方法
3 . 如图,已知球的表面积为
,
是该球的内接长方体(即该长方体的八个顶点均在球面上)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78b65065ec3a0cb4b050989165c003d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65fc998d33cdf37c272f79cfd64b7b99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
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解题方法
4 . 如图,在
中,
,斜边
是
的中点,现将
以直角边
为轴旋转一周得到一个圆锥,点
为圆锥底面圆周上的一点,且
.
(2)若某动点在圆锥侧面上运动,试求该动点从点
出发运动到点
所经过的最短距离;
(3)若一个棱长为
的正方体木块可以在这个圆锥内任意转动,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c633830c6e2ac6d8d6e18890ef5ee33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/991c8373be20b4325ba779e4dfdc8b15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7468a5b7fffe9d46e925874a866f6629.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c633830c6e2ac6d8d6e18890ef5ee33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2c3d2cba96f6f03520c0b3f6e4da03e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/886c25cbaff27a9c4cf52dacec0eac4c.png)
(2)若某动点在圆锥侧面上运动,试求该动点从点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
(3)若一个棱长为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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2023-11-14更新
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510次组卷
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4卷引用:上海市南汇中学2023-2024学年高二上学期期中数学试题
上海市南汇中学2023-2024学年高二上学期期中数学试题(已下线)8.3简单几何体的表面积与体积【第三练】“上好三节课,做好三套题“高中数学素养晋级之路(已下线)8.3.2 圆柱、圆锥、圆台、球的表面积和体积-同步精品课堂(人教A版2019必修第二册)福建省南安市蓝园高级中学2023-2024学年高一下学期期中考试数学试题
名校
5 . 已知圆锥的顶点为P,母线
所成角的余弦值为
,轴截面等腰三角形的顶角为
,若
的面积为
.
(1)求该圆锥的侧面积;
(2)求圆锥的内切球体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/790ef3382b1c731f2885eecfd92c2a86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d266a04f3dc7483eddbc26c5e487db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0468237bbc0d3df77435d98b817c10c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2205cffebf8c4d5f81d15ed7b85c8936.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e609d7f5a3b904e30f43fbbc26033d7.png)
(1)求该圆锥的侧面积;
(2)求圆锥的内切球体积.
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2023-11-13更新
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396次组卷
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5卷引用:江西省景德镇市昌江区景德镇一中2023-2024学年高二上学期11月期中考试数学试题
江西省景德镇市昌江区景德镇一中2023-2024学年高二上学期11月期中考试数学试题(已下线)考点7 组合体的内切 2024届高考数学考点总动员【练】(已下线)专题8.3 简单几何体的表面积与体积-举一反三系列(人教A版2019必修第二册)(已下线)高一下学期期中复习解答题压轴题十八大题型专练(2)-举一反三系列(人教A版2019必修第二册)山东省菏泽市菏泽一中系列2023-2024学年高一下学期4月期中考试数学试题(A)
6 . 我国古代数学名著《九章算术》,将底面为矩形且有一条侧棱垂直于底面的四棱锥称为“阳马”.如图所示,在长方体
中,已知
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/31/064926f8-580d-47cb-ba38-0fa73946e3aa.png?resizew=134)
(1)求证:四棱锥
是一个“阳马”,并求该“阳马”的体积;
(2)求该“阳马”
的外接球的表面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f121eabff3c62c1a196d9ca5f6f83f0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8d927585a17c2e98ef7d5a9589a26ac.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/31/064926f8-580d-47cb-ba38-0fa73946e3aa.png?resizew=134)
(1)求证:四棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fec35c2182c5e0c80b766adceb058e5f.png)
(2)求该“阳马”
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fec35c2182c5e0c80b766adceb058e5f.png)
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解题方法
7 . 已知任意三角形的三边长分别为
,内切圆半径为
,则此三角形的面积可表示为
.其原理是由内切圆的圆心与三角形三个顶点的连线把三角形分割成三个小三角形,每个小三角形的面积等于大三角形的边长与内切球半径的乘积的
,三个小三角形面积相加即得
.请运用类比思想,解决空间四面体中的以下问题.
(1)已知四面体四个面的面积分别为
,
,
,
,内切球的半径为
,请运用类比思想,写出该四面体的中的相应结论;
(2)应用(1)中的结论求解:已知三棱锥(又叫四面体)
,三条侧棱
,
,
两两垂直,且
,求此三棱锥的内切球半径.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53d69e0bbde9001538ffea1063d11db7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1c011c6b72ee4888607e272e2168178.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/13/ff37a84b-8751-4101-a6e8-7c7a4b05469a.png?resizew=147)
(1)已知四面体四个面的面积分别为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6899bf9cadae2ccdb14cbc87d4f280ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f30f56664446f32dbbc2c5f12a99374.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
(2)应用(1)中的结论求解:已知三棱锥(又叫四面体)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50094bfee564d9c1b03088ac2ece28c3.png)
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解题方法
8 . 在正三棱台
中,已知
,
,三棱台的高
.
(1)求棱台
的体积;
(2)若球
与正三棱台
内切(与棱台各面都相切),求球
的表面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f54638dd4ebf19815a1333d84e42f927.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5eb7962aeb7a76ea6089e87a9b08fbb6.png)
(1)求棱台
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
(2)若球
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
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2023-09-03更新
|
281次组卷
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5卷引用:浙江省七彩阳光新高考联盟2023-2024学年高二上学期返校联考数学试题
浙江省七彩阳光新高考联盟2023-2024学年高二上学期返校联考数学试题(已下线)考点7 组合体的内切 2024届高考数学考点总动员(已下线)第二章 立体几何中的计算 专题六 几何体的外接球、棱切球、内切球 微点18 几何体的内切球、棱切球综合训练【基础版】(已下线)专题13.8外接球与内切球3大题型13个方向-重难点突破及混淆易错规避(苏教版2019必修第二册)(已下线)专题13.6空间图形的表面积和体积-重难点突破及混淆易错规避(苏教版2019必修第二册)
解题方法
9 . 如图,在正四棱柱
中,
,
∥平面MAC.
(1)证明:M是
的中点;
(2)若正四棱柱的外接球的体积是
,求该正四棱柱的表面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92535536bd3c2761724fd058427f95a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fe734023d4e70010a6b2cc3267cb86e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/8/5/e7673eec-8fe4-47f5-a514-8bc8c82a4303.png?resizew=125)
(1)证明:M是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22adbc0da438220f9cace11b629d799b.png)
(2)若正四棱柱的外接球的体积是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eefdcd39676298f214654051fc51ba92.png)
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2023-07-28更新
|
571次组卷
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2卷引用:山东省青岛第十五中学2023-2024学年高二上学期期初考试数学试题
解题方法
10 . 如图,已知三棱锥
的三条侧棱
,
,
两两垂直,且
,
,
,三棱锥
的外接球半径
.
(1)求三棱锥
的侧面积
的最大值;
(2)若在底面
上,有一个小球由顶点
处开始随机沿底边自由滚动,每次滚动一条底边,滚向顶点
的概率为
,滚向顶点
的概率为
;当球在顶点
处时,滚向顶点
的概率为
,滚向顶点
的概率为
;当球在顶点
处时,滚向顶点
的概率为
,滚向顶点
的概率为
.若小球滚动3次,记球滚到顶点
处的次数为
,求数学期望
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00803e67a5d417a9a4dc00277fca778b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/495636df02b96acab4478baabe77bafa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69acd73890957b0007b30fd81f2abc0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e159fa38488741d395ea9cb03386b1ad.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/18/58b499d6-5736-43f6-94b9-dd6be5e9ef67.png?resizew=139)
(1)求三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
(2)若在底面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf31876698721a199c7c53c6b320aa86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf31876698721a199c7c53c6b320aa86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bf3baba074e8aeb6f3ea117865bbd1b.png)
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