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解题方法
1 . 《九章算术·商功》:“斜解立方,得两堑(qiàn)堵(dǔ).斜解堑堵,其一为阳马,一为鳖(biē)臑(nào).阳马居二,鳖臑居一,不易之率也.合两鳖臑三而一,验之以棊,其形露矣.”刘徽注:“此术臑者,背节也,或曰半阳马,其形有似鳖肘,故以名云·中破阳马,得两鳖臑,鳖臑之起数,数同而实据半,故云六而一即得.”阳马和鳖臑是我国古代对一些特殊锥体的称谓,取一长方体,按下图斜割一分为二,得两个一模一样的三棱柱,称为堑堵,再沿堑堵的一顶点与相对的棱剖开,得四棱锥和三棱锥各一个,以矩形为底,另有一棱与底面垂直的四棱锥,称为阳马,余下的三棱锥是由四个直角三角形组成的四面体,称为鳖臑.
:
①在右图中,求三棱锥
的高.
②求三棱锥
外接球的表面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c8cf5d11389d101e9ebf87764d0f8dd.png)
①在右图中,求三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac7a3e0ef4980cc0ca102f733d357263.png)
②求三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac7a3e0ef4980cc0ca102f733d357263.png)
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名校
解题方法
2 .
九章算术
商功
“斜解立方,得两堑
堵
斜解堑堵,其一为阳马,一为鳖
臑
阳马居二,鳖臑居一,不易之率也
合两鳖臑三而一,验之以棊,其形露矣
”刘徽注:“此术臑者,背节也,或曰半阳马,其形有似鳖肘,故以名云
中破阳马,得两鳖臑,鳖臑之起数,数同而实据半,故云六而一即得
”阳马和鳖臑是我国古代对一些特殊锥体的称谓,取一长方体,按下图斜割一分为二,得两个一模一样的三棱柱,称为堑堵
再沿堑堵的一顶点与相对的棱剖开,得四棱锥和三棱锥各一个.以矩形为底,另有一棱与底面垂直的四棱锥,称为阳马
余下的三棱锥是由四个直角三角形组成的四面体,称为鳖臑.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/31/2f4db74b-e7e3-4f58-a61b-90abce75befa.png?resizew=415)
(1)在下左图中画出阳马和鳖臑
不写过程,并用字母表示出来
,求阳马和鳖臑的体积比;
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/31/24b43e16-6ed4-4ea0-a84f-3aa49e073360.png?resizew=283)
(2)若
,
,在右图中,求三棱锥
的高.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33b0e787c1d82071c825975348698f58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e85bda46cc51c938224d9165301e3896.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93dd4ee75eaf5d8f2e1c758cb18a0341.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2c3cf2e7ed24dadc34e6216a3f5c4bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e48ef01e30a6ec3dd9940fd767030e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d589e7b70f38ec2f41b68c889a56482.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78be1239d7b3a80cead923442e1f8df5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c90282d4a37c9a20620d4bbb0c263cae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c90282d4a37c9a20620d4bbb0c263cae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c90282d4a37c9a20620d4bbb0c263cae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c90282d4a37c9a20620d4bbb0c263cae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c90282d4a37c9a20620d4bbb0c263cae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2db9a58e185e4fd9c4f86efb24480f1.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/31/2f4db74b-e7e3-4f58-a61b-90abce75befa.png?resizew=415)
(1)在下左图中画出阳马和鳖臑
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd995178601c2ad7b40f973d268c7bb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04582116cd765fcc5a52f44279ad6c94.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/31/24b43e16-6ed4-4ea0-a84f-3aa49e073360.png?resizew=283)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3813fa868b9a107058dc709145746437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef0402dd5ae3db10281f9f1e11738bcb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9651b2b082f5de09e5a410804c4c2c0f.png)
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