Volume of a tetrahedron regular

In summary, the conversation discusses writing the surface and volume of a tetrahedron in terms of different variables such as x, y, z, u, v, w, A, B, and C. It is mentioned that there are complicated formulas for computing the volume, and a link is provided for further study. The conversation also mentions the possibility of writing the volume in terms of the areas A, B, C, and S, and a solution is given for writing the volume in terms of u, v, and w. Finally, it is stated that the volume can also be written in terms of A, B, and C, and a solution is provided for that as well.
  • #1
Bruno Tolentino
97
0
See the image that I uploaded...

2222.png


I want to write the surface S (bounded by edges u, v and w) in terms of x, y and z, u, v and w and A, B and C. And I got it!

See:
[tex]S(A,B,C) = \sqrt{A^2+B^2+C^2}[/tex]
[tex]S(x,y,z) = \sqrt{\frac{1}{4}( (yz)^2 + (zx)^2 + (xy)^2 )}[/tex]
[tex]S(u,v,w) = \sqrt{(+u+v+w)(-u+v+w)(+u-v+w)(+u+v-w)}[/tex]

And the Volume V:
[tex]V(x,y,z) = \frac{1}{6} xyz [/tex]
But, I don't know how to write V in terms of A, B, C neither u, v, w. Can you help me with this, please?
 
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  • #2
Bruno Tolentino said:
See the image that I uploaded...

View attachment 84788

I want to write the surface S (bounded by edges u, v and w) in terms of x, y and z, u, v and w and A, B and C. And I got it!

See:
[tex]S(A,B,C) = \sqrt{A^2+B^2+C^2}[/tex]
[tex]S(x,y,z) = \sqrt{\frac{1}{4}( (yz)^2 + (zx)^2 + (xy)^2 )}[/tex]
[tex]S(u,v,w) = \sqrt{(+u+v+w)(-u+v+w)(+u-v+w)(+u+v-w)}[/tex]

And the Volume V:
[tex]V(x,y,z) = \frac{1}{6} xyz [/tex]
But, I don't know how to write V in terms of A, B, C neither u, v, w. Can you help me with this, please?
There are some complicated Heron-type formulas for computing the volume of a tetrahedron which are similar to those for computing the area of a triangle. However, these formulas are much more complex.

The paper at the following link shows the derivation of these formulas ad gives references for further study:

http://www.cs.berkeley.edu/~wkahan/VtetLang.pdf

Tetrahedrons are discussed starting at p. 11, but the previous material provides a good refresher.
 
  • #3
I thank you for this answer. Actually, this no answer my question, but I'll intend to ask this in another thread. I'll intend to ask this and more one thing, that's the following:

Given a tetrahedron irregular (any tetrahedron), how to write the volume V in terms of the areas A, B, C and S?

OBS: my first question in this thread still no be answered.
 
  • #4
Bruno Tolentino said:
I thank you for this answer. Actually, this no answer my question, but I'll intend to ask this in another thread. I'll intend to ask this and more one thing, that's the following:

Given a tetrahedron irregular (any tetrahedron), how to write the volume V in terms of the areas A, B, C and S?

OBS: my first question in this thread still no be answered.

If the edges x, y, and z are mutually perpendicular, you can write expressions for the areas A, B, and C using those lengths.
 
  • #5
SteamKing said:
If the edges x, y, and z are mutually perpendicular, you can write expressions for the areas A, B, and C using those lengths.

I don't understand you explanation...

I discovered how to write V in terms of u, v and w:

[tex]V(u,v,w) = \sqrt{\frac{1}{288} (+u^2+v^2-w^2) (+u^2-v^2+w^2) (-u^2+v^2+w^2) }[/tex]
 
  • #6
I discovered too: [tex]V(A,B,C) = \sqrt{\frac{2}{9} A B C} [/tex]
 

1. What is the formula for calculating the volume of a regular tetrahedron?

The formula for calculating the volume of a regular tetrahedron is V = (a³ * √2) / 12, where "a" represents the length of one edge of the tetrahedron.

2. What is a regular tetrahedron and how is it different from other tetrahedrons?

A regular tetrahedron is a three-dimensional shape with four triangular faces, all of which are equilateral triangles. It is different from other tetrahedrons because all of its faces are identical in size and shape, and all of its angles are equal.

3. How do I measure the length of an edge on a regular tetrahedron?

To measure the length of an edge on a regular tetrahedron, simply measure the distance from one vertex to the opposite vertex. Since all edges on a regular tetrahedron are equal in length, you can measure any edge and it will be the same length as all others.

4. Can the volume of a regular tetrahedron be calculated using any unit of measurement?

Yes, the volume of a regular tetrahedron can be calculated using any unit of measurement as long as all measurements (such as length of edge) are in the same unit. It is important to be consistent with units when calculating volume.

5. How is the volume of a regular tetrahedron related to its surface area?

The volume of a regular tetrahedron is directly related to its surface area. The volume is equal to one third of the product of the surface area and the distance between any two opposite faces. This means that as the surface area increases, the volume also increases.

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