Volume of a tetraedron in function of the areas

In summary, the conversation discusses calculating the volume of a tetrahedron using the areas of its surfaces. The PDF provided explains that in addition to the 4 faces, 3 pseudo-faces are necessary to accurately calculate the volume. The question is raised if it is possible to calculate the volume with just the 4 faces. The issue is that the area of a triangle does not determine its shape, making it difficult to accurately calculate the volume with only 4 surface areas.
  • #1
Bruno Tolentino
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Given any tetraedron, I want to calculate the volume V in function of the areas of the surfaces of the solid. I found this pdf that explain this:

http://daylateanddollarshort.com/mathdocs/Heron-like-Results-for-Tetrahedral-Volume.pdf

But, o pdf says that beyond of the 4 faces (X, Y, Z, W) is necessary more 3 pseudo-faces (H, J, K). But, is it correct? Is really necessary 7 areas for compute the volume? With just 4 is not possible?
 
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  • #2
Have you tried working it out for the areas of 4 faces alone?

The trouble is that the area of a triangle does not determine it's shape - so you can construct many differently volumed tetrahedra out of four triangles knowing only their areas (but not their shapes).
 
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Related to Volume of a tetraedron in function of the areas

What is the formula for calculating the volume of a tetrahedron?

The formula for calculating the volume of a tetrahedron is V = (1/3) * A * h, where A is the area of any of the four triangular faces and h is the height of the tetrahedron.

How do you find the area of a triangular face in a tetrahedron?

To find the area of a triangular face in a tetrahedron, you can use the formula A = (1/2) * b * h, where b is the base of the triangle and h is the height of the triangle. The base can be found by dividing the length of any edge by 2, and the height can be calculated using the Pythagorean theorem.

Can the volume of a tetrahedron be negative?

No, the volume of a tetrahedron cannot be negative. It is a measure of the amount of space enclosed by the tetrahedron and therefore must be a positive value.

Is there a relationship between the volume and area of a tetrahedron?

Yes, there is a relationship between the volume and area of a tetrahedron. As the area of the triangular faces increases, the volume of the tetrahedron also increases. This is because a larger area means a larger amount of space is enclosed by the tetrahedron.

Can the volume of a tetrahedron be calculated without knowing the height?

Yes, the volume of a tetrahedron can be calculated without knowing the height. You can use the formula V = (1/3) * A * h, where A is the area of any of the triangular faces. However, in order to calculate the height, you will need to know the length of at least one of the edges of the tetrahedron.

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