名校
解题方法
1 . 在边长为4的正方形ABCD中,如图甲所示,E,F,M分别为BC,CD,BE的中点,分别沿AE,AF及EF所在直线把
和
折起,使B,C,D三点重合于点P,得到三棱锥
,如图乙所示,则三棱锥
外接球的体积是____________ ;过点M的平面截三棱锥
外接球所得截面的面积的取值范围是____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/271bed6ec1c0206b165dec255f8f0bb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f5a0a6e5b3f489a7032ea5116c96024.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b115316e0fcd2ef46a4dd383472996e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b115316e0fcd2ef46a4dd383472996e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b115316e0fcd2ef46a4dd383472996e4.png)
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2 . 在平面五边形
中,
,
,
,
,且
.将五边形
沿对角线
折起,使平面
与平面
所成的二面角为
,则沿对角线
折起后所得几何体的外接球的体积为_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9142a8490de14a87eda628ffa7e28982.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5b8b98b2f83279a49e94d9f48c5e6f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9c14f5c9eafaeff843e4aa4c18ecce1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4bd6a2b112facda441f4e34bf5c145fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8e41916523511064a97de39b0f2b323.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f6c2faabb08a3712336f812b0c69db4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9142a8490de14a87eda628ffa7e28982.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c09afc70f448545336304333d5b5658b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fa7bbd7831e9ff4f8cffc8889d34f05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
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3 . 已知
是球
表面上的点,
平面
若球
的体积为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c3e0e93b586844f67ca7a3b157dd310.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/523bdb05d8e5de2a84ccedb6db738037.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10c83f8945042b9c8fb2fbdac9308d62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c880085b1eb986f6ce3653337433789.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67fb457e8ac0d3ac35e1c668ea138f91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c3e0e93b586844f67ca7a3b157dd310.png)
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4 . 在《九章算术》中,将四个面都是直角三角形的四面体称为鳖臑,在鳖臑
中
,
,
,
,
,已知动点E从C点出发,沿外表面经过棱AD上一点到点B的最短距离为
,则该棱锥的外接球的体积为________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/891579e7c231584a8e16b8eeff79888e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/080db3af81b29ed10144a1c2e2a4fb8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8915e8e775538d41debf1933102c6b86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c90da0cd2708481057fe19acebf2ec7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce4ab7e657f01bdfa235f8c4d6681d13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ed18447a59016c8c89d1561f7dd5172.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4056761b8f826eeb6ad8c9a151d3c9c.png)
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5 . 中国古代数学著作《九章算术》中,将底面是直角三角形的直三棱柱称之为“堑堵”,将底面为矩形,一条侧棱垂直于底面的四棱锥称之为“阳马”.在如图所示的堑堵
中,
,则阳马
的外接球的体积与表面积之比是__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cac2e4804b5efbe1e32540462334f600.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895ac202e3507cb633337b41299ad84b.png)
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6 . 已知边长为2的等边
中,
为
的中点,以
为折痕进行折叠,使折后的
,则过
四点的球的体积为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1287090703aac6d26361c4212862bcb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c82a10b4f0c9323d726804c89dd9548.png)
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7 . 我国魏晋时期的数学家刘徽(图a)创造了一个称为“牟合方盖”的立体图形,在正方体内作两个互相垂直的内切圆柱(图b),其相交的部外就是牟合方盖(图c).我国南北朝时期数学家祖暅基于“势幂既同则积不容异”这一观点和对牟合方盖性质的研究,推导出了球体体积公式.已知在一个棱长为2r的正方体内有一个牟合方盖(图1),设平行于水平面且与水平面距离为
的平面为
,则平面
截牟合方盖所得截面的形状为__________ (填“正方形”或“圆形”),设这个牟合方盖的体积为
(图2),并设半径为
的球的体积为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/caa4c480d031dedac6e81872836d04cc.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5547c2bb8607c1dba2bb0881777dbb34.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4764374bd2fb78e59cd0b283637baeb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63055a5d6916f99d07fede49120753f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/caa4c480d031dedac6e81872836d04cc.png)
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8 . 如图所示,三棱锥
中,平面
平面ABC,
,
,
,
,则三棱锥
外接球的体积为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d077f6da8b2c00b152d4679aa2ed7f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/036de574712cad14bddadf6653c7e714.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/615fc8790237a1b09af51d6bcad6b595.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c24b7a9466a1e35328a8a4b1ba7fa84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b10134e7a46e6f6f7cb9d5e2371727d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
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9 . 在棱长为4的正方体
中,点
是棱
的中点,则四面体
的外接球的体积为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0a851907ada2ac2c3c4880a6736d28a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c27ae22ad74b412e6d71ec1245f802db.png)
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10 . 如图是四棱锥
的平面展开图,四边形
是矩形,
.在四棱锥
中,M为棱PB上一点(不含端点),则下列说法正确的是__________ .
的最小值为
;
②存在点M,使得
;
③四棱锥
外接球的体积为
;
④三棱锥
的体积等于三棱锥
的体积
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2e0a20c9616658eadb94c25e93b3052.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/481b196bed2a1500dee6722f1525b158.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/741642aa6f98d6bd11b3a4d8212d811b.png)
②存在点M,使得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07789d6a8259ac89144caa816aaaf47d.png)
③四棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8b0a70e166de9359b4a1d0854286049.png)
④三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e682db81a82443f63a567eb29f4aa7bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c653b8894344786624fd44bfd636d6b.png)
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2卷引用:四川省峨眉市第二中学校2024届高三适应性考试暨押题数学(文)试题