1 . 祖暅在求球体积时,使用一个原理:“幂势既同,则积不容异”.“幂”是截面积,“势”是立体的高.意思是两个同高的立体,如在等高处的截面积恒相等,则体积相等.更详细点说就是,界于两个平行平面之间的两个立体,被任一平行于这两个平面的平面所截,如果两个截面的面积恒相等,则这两个立体的体积相等.上述原理在中国被称为祖暅原理.如图是一个半径为
的球体,平面
与球相交,截面为圆
,延长
,交球于点
,则
垂直于圆
(
垂直于圆
内的所有直线).
,求圆锥DB的表面积和体积;
(2)如图平面
上方与球体之间的部分叫球冠,若
,请你利用祖暅原理求球冠的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaf1438142deeac876fc7dc50552e552.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaf1438142deeac876fc7dc50552e552.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaf1438142deeac876fc7dc50552e552.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0be901bc31043f90194eb4131bf22b00.png)
(2)如图平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a6c9414983f9771bbb06bf9a3e75f8d.png)
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解题方法
2 . 如图,在直三棱柱
中,
,
,
、
分别为
、
的中点,设平面
交棱
于点
.
;
(2)求二面角
的平面角的正切值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/080db3af81b29ed10144a1c2e2a4fb8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a1ef1ea57ed911896ff5568c9998ac2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0a851907ada2ac2c3c4880a6736d28a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/657dffbd3623b705f871878fbd9df57e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12e69f1bfd13ddd15449f75343c221c9.png)
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3 . 如图,在三棱锥
中,
,
,
,
,点
在
上,点
为
的中点.
平面
;
(2)求
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4181847b09918a22b8230529d0c5bb8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/364d6c88726d8c3bb8ed297057332bac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb1e4bb4231e0e8fc02cdc794a6c491c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b5f215a42c4b7078d8d65923eb9980e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/764509115979e9958101808383672ec0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e61620a272dada8d4b9a9fab6379dfd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93edc7bb513f40a89173121c8570cd65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/840798a31aba0783f96584e0ad7c0d2e.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0213c5787a5a6b38d11bceca5567f67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/840798a31aba0783f96584e0ad7c0d2e.png)
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解题方法
4 . 《九章算术》作为中国古代数学专著之一,在其“商功”篇内记载:“斜解立方,得两堑堵.斜解堑堵,其一为阳马,一为鳖臑”,鳖臑是我国古代数学对四个面均为直角三角形的四面体的统称.在长方体
中,已知
.
平面
;
(2)求
与平面
所成角的正弦值;
(3)求平面
与平面
夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38c014c8923b1ce9dcb8b028dd8b9f0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61cdaadeae37736a1e6dd93fa1fe712f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4508dc6d9c91157836be679c0543cac.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7f6f93171329d508d491143b9d71f7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc4fd5b13f66aaa25632811704596c44.png)
(3)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9afac7c616bbb14e1ed428a3c507c7dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4508dc6d9c91157836be679c0543cac.png)
您最近一年使用:0次
5 . 如图,三棱台
中,
是边长为2的等边三角形,四边形
是等腰梯形,且
,
为
的中点.
;
(2)若过
三点的平面截三棱台
所得的截面面积为
.当二面角![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbf41a3fcbd83bb806c9c4cbec8e36d2.png)
为锐二面角时,求二面角
的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d9a8181f7a7fe7f3fac872ce9534f15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51f856c50b63de74c8e85c608e9dcc0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f1f229274a6e17977cc047814212589.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfc1f76257275ab4b04f9bc913535670.png)
(2)若过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb7ca1a5419f8a52b3141b0bc7b47dad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d3eb19246fe0299ae396fc9a7e1ee6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbf41a3fcbd83bb806c9c4cbec8e36d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e07eba786740939ce0ec951572b02f0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047b9fd4020265766074c80a7f8ad3a0.png)
您最近一年使用:0次
6 . 如图,在棱长为1的正四面体
中,
是
的中点,
,
分别在棱
和
上(不含端点),且
平面
.
平面
;
(2)若
为
中点,求平面
截该正四面体所得截面的面积;
(3)当直线
与平面
所成角为
时,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/891579e7c231584a8e16b8eeff79888e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63a779876cdfb2c489ad0eaed0f73e6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f81fa367ec317fe2a30142e1c30cce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffe8a84ca3a13f82aff1a022edc66065.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffe8a84ca3a13f82aff1a022edc66065.png)
(3)当直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e55e398e8520d8a36fb5a625a085b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca67a5b8f69507c8b80379e86f90a8ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/037fb348109dc2063a268b10eb925a57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e72b2e1ff83e95df048745322982451.png)
您最近一年使用:0次
解题方法
7 . 如图,在四棱锥
中,底面
是边长为1的正方形,
底面
,
,
是线段
的中点.
平面
;
(2)求三棱锥
的体积;
(3)求直线
与底面
所成角的正切值.
![](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/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b80ee363635d73f601654339028daec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa69a2247ad4d5231aa361349b12f97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53bdef2e7a7929ad6190302ab44c46c0.png)
(2)求三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9bad875ab4b5b8c707d452db4cabaa4.png)
(3)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69d2b798744645af88a4fa411344a83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
您最近一年使用:0次
解题方法
8 . 如图,直线
,
,
相交于点
,
,
,
,
,
,
.
(1)求证:平面
平面
;
(2)
为
中点,求
与平面
所成角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/018af6682b038cfdf6ed136943413271.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ed7bc83ebd009c5a7f58116eb8f5d8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbd93ad139410a0bc761cce65c84f599.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77afdfddf403092baca86673027f595a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29d07d4bc0fc8bb2c1e5cf8222d40be9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0386390c526190b3109295ef7447fc2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0fca3f37114714d0f9f0d2c283c3718.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c624edeecdc4703f55d28f01f2d5b46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e65a3e478bb87d094e3a0af30dd10ae8.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/6/21/1df60921-1a93-4ab5-bf65-6a8241e0688d.png?resizew=147)
(1)求证:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3d7090639341730951c1bc3c9b6164e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ed01d1ff5a7f21a68fb3a1e5c7f393e.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a85c632e08ce1b355808bc7a9ad9a051.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
您最近一年使用:0次
名校
9 . 如图,在四棱锥
中,底面
为正方形,侧面
底面
,且
.
平面
;
(2)求三棱锥
的体积;
(3)求二面角的
正切值.
![](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/93edc7bb513f40a89173121c8570cd65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7157fb6bcd229e82079a471898ab438.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d0edb1508fc95765f3bb316bcb5252d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
(2)求三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/041434f0c90fb3cdd685b8eb1c2b4b26.png)
(3)求二面角的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70682f6196e6c1a08eb48da73e8919ca.png)
您最近一年使用:0次
名校
10 . 如图,在直三棱柱
中,
,
,四边形
为正方形.
平面
;
(2)求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27031fbda748bc03320346e7c4d26fe1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71807a35b3170fce28ee6edf4c00d083.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f96c673a2381f118ea2d3efc0bca1f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a99ef32b30524326ce26f117cd7f5a91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f96c673a2381f118ea2d3efc0bca1f3.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/966b2d5659b3dc130fe0e4b2c0ff0072.png)
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3卷引用:浙江省重点中学四校2023-2024学年高一下学期5月联考数学试题
浙江省重点中学四校2023-2024学年高一下学期5月联考数学试题 山东省济宁市第一中学2023-2024学年高一下学期6月月考数学试题(已下线)专题08 立体几何大题常考题型归类-期末考点大串讲(人教B版2019必修第四册)