Volume（頂級難）

doraemonpaul

2007-03-31 4:05 am

There is a 三角錐體 ABCD where AB = 11 cm, AC = 12 cm, AD = 13 cm, BC = 14 cm, BD = 15 cm and CD = 16 cm. Find its volume.

The answer should be in exact value.

回答 (2)

魏王將張遼

2007-04-01 7:29 am

✔ 最佳答案

We may use integration of piecewise volume of triangular plates as follows:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Maths/Doraemonapril1.jpg

Please note that △EFG is similar to △ABC if:
DE/DA = DF/DB = DG/DC
So viewing each of the side planes of the irregular tetrahedron, we have:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Maths/Doraemonapril2.jpg

And therefore, area of △EFG (and hence its volume) can be found by Heron's formula as follows:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Maths/Doraemonapril3.jpg

Finally, integrating from base to the vertex (i.e. x from 0 to 13), we have:

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Maths/Doraemonapril4.jpg

參考: My Maths knowledge

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siu mui

2007-04-01 5:07 am

首先先設底面面積的高BE,然后解得某個數,根據三垂綫定理可以知度ABE係RT三角形,再解得高,再代入V=(1/3)X底面積X高

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