名校
解题方法
1 . 如图一:球面上的任意两个与球心不在同一条直线上的点和球心确定一个平面,该平面与球相交的图形称为球的大圆,任意两点都可以用大圆上的劣弧进行连接.过球面一点的两个大圆弧,分别在弧所在的两个半圆内作公共直径的垂线,两条垂线的夹角称为这两个弧的夹角.如图二:现给出球面上三个点,其任意两个不与球心共线,将它们两两用大圆上的劣弧连起来的封闭图形称为球面三角形.两点间的弧长定义为球面三角形的边长,两个弧的夹角定义为球面三角形的角.现设图二球面三角形
的三边长为
,
,
,三个角大小为
,
,
,球的半径为
.![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf538440bd45e5881f2b22994560ba7a.png)
(2)①求球面三角形
的面积
(用
,
,
,
表示).
②证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f435efcc7869eec21bdba1ed81dc3f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf538440bd45e5881f2b22994560ba7a.png)
(2)①求球面三角形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f435efcc7869eec21bdba1ed81dc3f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
②证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f314e3f1d6311f0476623d4e55484a3e.png)
您最近一年使用:0次
2023-04-21更新
|
384次组卷
|
4卷引用:浙江省A9协作体2022-2023学年高一下学期期中联考数学试题
浙江省A9协作体2022-2023学年高一下学期期中联考数学试题(已下线)13.3 空间图形的表面积和体积(分层练习)江苏省徐州市第一中学2022-2023学年高一下学期期中数学试题(已下线)11.1.5 旋转体-【帮课堂】(人教B版2019必修第四册)
2 . 在
中,角
的对边分别是
,若
,
.
(1)证明:
是正三角形.
(2)若
的三顶点都在球O表面,且球O的表面积为
,求三棱锥
的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4e41e7cd5514d3ac0e7adc537882317.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/711232b7b300f1c5fbc07c978a4c7bb5.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4a39dce3f1e36dbe01293c309816968.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4278c0911e7df78965e78cff69cac5f5.png)
您最近一年使用:0次
21-22高一下·浙江·期中
名校
3 . 在直三棱柱
中,
,
,
,D是AB的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/3/8d8bf36c-215e-4425-a5f2-85b079e045a5.png?resizew=200)
(1)求三棱锥
的体积;
(2)求证:
∥平面
;
(3)求三棱柱
的外接球的表面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d70dc2c20619a4fc12a0cfda59af5b69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e65a3e478bb87d094e3a0af30dd10ae8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef01b27051dd18c0041e06406e12ef40.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/3/8d8bf36c-215e-4425-a5f2-85b079e045a5.png?resizew=200)
(1)求三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3a7f2b719a8ff2de7883ec2f2c27731.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24bb49fdc6b6bbb2449fdf8a0de769d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbc7e774e4ae40c23bf4ceed179230ca.png)
(3)求三棱柱
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
您最近一年使用:0次
2022-09-29更新
|
1444次组卷
|
3卷引用:高中数学 高一下-7
20-21高一下·浙江·期末
名校
4 . 如图,在棱长为
的正方体
中,
点是
的中点.
![](https://img.xkw.com/dksih/QBM/2021/6/2/2734570351747072/2734863987220480/STEM/a76772f9-52d8-4949-bd00-06a17d7c292d.png?resizew=211)
(1)证明:
平面
;
(2)求三棱锥
外接球的表面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](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/22adbc0da438220f9cace11b629d799b.png)
![](https://img.xkw.com/dksih/QBM/2021/6/2/2734570351747072/2734863987220480/STEM/a76772f9-52d8-4949-bd00-06a17d7c292d.png?resizew=211)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60426d3c6f8c8bde775914fa9f0a7fe0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81bf7a859123936a07193592e089340a.png)
(2)求三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61df721e926fddc37c15a341e4559a28.png)
您最近一年使用:0次
2021-06-03更新
|
1379次组卷
|
3卷引用:【新东方】高中数学20210527-021【2021】【高一下】
(已下线)【新东方】高中数学20210527-021【2021】【高一下】新疆维吾尔自治区2022-2023学年高一下学期期末考试数学试题山西省阳泉市第一中学校2023-2024学年高二上学期开学分班数学试题
名校
5 . 设三棱锥
的每个顶点都在球
的球面上,
是面积为
的等边三角形,
,
,且平面
平面
.
![](https://img.xkw.com/dksih/QBM/2020/1/15/2377714267447296/2379173098070016/STEM/1ba171fefa4f418ead367b327a00f4dd.png?resizew=207)
(1)求球
的表面积;
(2)证明:平面
平面
,且平面
平面
.
(3)与侧面
平行的平面
与棱
,
,
分别交于
,
,
,求四面体
的体积的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e43f6eb62e48e72e06361138e0d1e4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/701f3b0e2bedfe5195443459072d798e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4c8fb96739aafdf34bc1f98f6340221.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92863f6b0a1aaecf38b68b3c9e26f496.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea890bf7c94c507185d6d9171299d312.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03cc9c1600db3a5653e5903db8286e43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51cc198525fa3652d468c5c74dae8c26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://img.xkw.com/dksih/QBM/2020/1/15/2377714267447296/2379173098070016/STEM/1ba171fefa4f418ead367b327a00f4dd.png?resizew=207)
(1)求球
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/701f3b0e2bedfe5195443459072d798e.png)
(2)证明:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ddd943015f1d02a2b3706ccef0e6ac2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ddd943015f1d02a2b3706ccef0e6ac2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
(3)与侧面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07c9121cede0ee0562e23b8a26b34616.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/289d7a880379d6060065c829b45b0ed6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/252053b853152bd294a8315debd00b92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/655b6a742dbacdab5aaa298007663dd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c10f93679abcee21bacd92c3b1552a0e.png)
您最近一年使用:0次
2020-01-17更新
|
461次组卷
|
5卷引用:思想01 函数与方程思想 第三篇 思想方法篇(讲) 2021年高考二轮复习讲练测(浙江专用)
(已下线)思想01 函数与方程思想 第三篇 思想方法篇(讲) 2021年高考二轮复习讲练测(浙江专用)(已下线)思想03 数形结合思想 第三篇 思想方法篇(练)-2021年高考二轮复习讲练测(浙江专用)2020年1月广东省大联考高三数学(文科)试题云南省普洱市景东彝族自治县第一中学2019-2020学年高一月考数学试题(已下线)思想01 函数与方程思想(练)--第三篇 思想方法篇-《2022年高考数学二轮复习讲练测(新高考·全国卷)》
10-11高三下·浙江杭州·阶段练习
6 . 在直角梯形
中,
,
,且
分别是边
上的一点,沿线段
分别将
翻折上去恰好使
重合于一点![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/14/f20f29de-965d-46d3-aee6-6ded816cf479.png?resizew=245)
(I)求证:
;
(II)已知
,
,试求:
(1)四面体
内切球的表面积;
(2)二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60add450a5ddb5b0f3cce4a560d2d0a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cff860cac2f40cd3311d1d7baa314ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9247f5ae37d8fe6f2f404e1f921d64c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab609a6574633ebabcff3e73fa862081.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5cb72419cdf0534283b7e54194ac392.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52320bd8186f11ba9f8bb995ce7a339f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e788eaaad9ac3b3642f9914040e5dc5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6fa157b4f65f3a9aa1f7f82de02e99e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/14/f20f29de-965d-46d3-aee6-6ded816cf479.png?resizew=245)
(I)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a77e3c1c236141d6118429fade0a9b9d.png)
(II)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7a68ccad27fc7fc7ab70362a35d2dd0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c51249f1ff811c35abbc8f6dc5a8044e.png)
(1)四面体
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(2)二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ec2524be492bca0d1566bf848066f10.png)
您最近一年使用:0次