Calculate Volume of Tetrahedron with Given Vertices | Step-by-Step Solution

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Homework Help Overview

The problem involves calculating the volume of a tetrahedron defined by specific vertices in three-dimensional space. The vertices given are (0,0,0), (1,0,0), (0,1,0), and (0,0,1). Participants are exploring the correct method for determining the volume and discussing the validity of the original poster's calculations.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • The original poster attempts to use a triple integral to find the volume and questions the correctness of their result. Some participants suggest using a volume formula for tetrahedrons instead. There is also a discussion about the geometric interpretation of the vertices and whether they indeed form a tetrahedron.

Discussion Status

Participants are actively engaging with the problem, with some providing corrections and clarifications regarding the volume calculation. There are differing opinions on the interpretation of the vertices and the resulting volume, indicating a productive exploration of the topic.

Contextual Notes

Some participants question the original poster's understanding of the geometric configuration of the vertices, suggesting that it may not represent a tetrahedron as initially thought. This has led to a discussion about the implications of the shape on the volume calculation.

madachi
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Homework Statement



Find the volume of the tetrahedron with vertices at (0,0,0),(1,0,0),(0,1,0),(0,0,1)

The Attempt at a Solution



I worked out the triple integral and found out that the volume is \frac{1}{6}? Is this correct? I know there is probably a much quicker way working the volume by just using the volume formula for tetrahedron. However, I am not sure which value to substitute to the formula, so could you just tell me whether this answer is right or not?

Thanks!
 
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The vertices you give do not make a tetrahedron. Try to draw the points in three dimensions. You'll see the volume is just half that of a cube with length 1. So the volume you're seeking should be 1/2.
 
For a tetrahedron, like a cone, the area is (1/3)*(area of the base)*height. So (1/3)*(1/2)*1=1/6, yes.
 
Mr. Miyagi is wrong. That is in fact a tetrahedron, it is 1/6 of a cube, not 1/2, and its volume is, indeed, 1/6.

More generally, the volume of the tetrahedron is with vertices at (0, 0, 0), (a, 0, 0), (0, b, 0), and (0, 0, c) is (abc)/6.
 
Ugh, sorry about that... Is it too late to claim temporary insanity? :rolleyes:

Thanks for correcting it so quickly.
 

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