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| Title: | Mathematics at DEC |
|
| Moderator: | RUSURE::EDP |
|
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
385.0. "Volume of Tetrahedron" by TOOLS::STAN () Thu Nov 21 1985 00:16
There are several problems in this note file that involve the
volume of a tetrahedron. Since formulas for the volume of a
tetrahedron are not too widely known, I reproduce some useful
information here.
If the coordinates of the vertices of a tetrahedron are
(0,0,0), (x1,y1,z1), (x2,y2,z2), (x3,y3,z3), then the
volume, V, of the tetrahedron is given by the equation:
| x1 y1 z1 |
| |
6 V = | x2 y2 z2 |
| |
| x3 y3 z3 |
If the lengths of the sides of the base of the tetrahedron are a, b, and c,
and the lengths of the edges opposite these sides are A, B, and C,
respectively, then the volume, V, is given by the equation:
| 2 2 2 |
| 0 A B C 1 |
| |
| 2 2 2 |
| A 0 c b 1 |
2 | |
288 V = | 2 2 2 | .
| B c 0 a 1 |
| |
| 2 2 2 |
| C b a 0 1 |
| |
| 1 1 1 1 0 |
2
Expanded out, this is 288 V =
2 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 2
- C c - a b c + C b c + B b c + C a c + A a c - C c
2 2 2 2 2 2 2 2 2 2 4 2 2 2 2 2 2 2 2 2
+ B C c + A C c - A B c - B b + B a b + A a b + B C b
2 2 2 4 2 2 2 2 2 4 2 2 2 2 2 2 2 2 2 4 2
- A C b - B b + A B b - A a - B C a + A C a + A B a - A a
This can also be written in the following form:
| 2 2 2 2 2 2 2 |
| 2 A - c + B + A - b + C + A |
2 | |
288 V = | 2 2 2 2 2 2 2 |
| - c + B + A 2 B - a + C + B | .
| |
| 2 2 2 2 2 2 2 |
| - b + C + A - a + C + B 2 C |
A tetrahedron is called isosceles, if its opposite edges are equal in length
(i.e. A=a, B=b, C=c). In that case, the volume is given by:
2 2 2 2 2 2 2 2 2 2
72 V = (a + b - c ) (b + c - a ) (c + a - b ) .
The tetrahedron is called regular, if all its edges have the same length, s.
In that case, the volume is given by:
3
V = s / 6 sqrt(2) .
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