Volume of a Tetrahedron: General Slicing Method

In summary, the general slicing method to find the volume of a tetrahedron is given by dividing the base into 3 congruent triangles and using the apothem and slant height to determine the height of the tetrahedron. Cross sections perpendicular to the $x$-axis will have a linear function for $s(x)$, which can be used to calculate the volume of each slice and then summed up using integration. The final formula for the volume is $\frac{\sqrt{2}s^3}{12}$, where $s$ is the edge length of the tetrahedron.
  • #1
MarkFL
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Here is the question:

General slicing method to find volume of a tetrahedron?


General slicing method to find volume of a tetrahedron (pyramid with four triangular faces), all whose edges have length 6?

I have posted a link there to this thread so the OP can view my work.
 
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  • #2
Hello Bringing truth.

Let's orient our coordinate axis, the $x$-axis, such that it is perpendicular to the base and passes through the apex. Let the point where this axis intersects the base be the origin of the axis. The distance from the origin to the apex along this axis is the height $h$ of the tetrahedron. To determine $h$, we may construct a right triangle whose shorter leg is the apothem $a$ of the base, the longer leg is $h$, and the hypotenuse is the slant height $\ell$ of one of the adjoining faces. Let $s$ be the edge lengths os the tetrahedron.

Let's first determine the apothem $a$. If we divide the base into 3 congruent triangles, we may equate the area of the whole triangle to the 3 smaller triangles:

\(\displaystyle \frac{1}{2}s^2\frac{\sqrt{3}}{2}=3\cdot\frac{1}{2}sa\)

\(\displaystyle s\frac{\sqrt{3}}{2}=3a\)

\(\displaystyle a=\frac{s}{2\sqrt{3}}\)

And the slant height is given by:

\(\displaystyle \ell=\frac{\sqrt{3}}{2}s\)

Hence:

\(\displaystyle h=\sqrt{\left(\frac{\sqrt{3}}{2}s \right)^2-\left(\frac{s}{2\sqrt{3}} \right)^2}=s\sqrt{\frac{3}{4}-\frac{1}{12}}=s\sqrt{\frac{2}{3}}\)

Now, if we take cross sections perpendicular to the $x$-axis, we may give the volume of an arbitrary slice as:

\(\displaystyle dV=\frac{\sqrt{3}}{4}s^2(x)\,dx\)

We know $s(x)$ will be a linear function, containing the points:

\(\displaystyle (0,s)\) and \(\displaystyle (h,0)\)

Thus the slope of the line is:

\(\displaystyle m=-\frac{s}{h}\)

And thus, using the point-slope formula, we find:

\(\displaystyle s(x)=-\frac{s}{h}x+s=\frac{s}{h}\left(h-x \right)\)

Hence, we may state:

\(\displaystyle dV=\frac{\sqrt{3}s^2}{4h^2}\left(h-x \right)^2\,dx\)

Summing up the slices, we may write:

\(\displaystyle V=\frac{\sqrt{3}s^2}{4h^2}\int_0^h\left(h-x \right)^2\,dx\)

Using the substitution:

\(\displaystyle u=h-x\,\therefore\,du=-dx\) we obtain:

\(\displaystyle V=-\frac{\sqrt{3}s^2}{4h^2}\int_h^0 u^2\,du=\frac{\sqrt{3}s^2}{4h^2}\int_0^h u^2\,du\)

Applying the FTOC, we have:

\(\displaystyle V=\frac{\sqrt{3}s^2}{12h^2}\left[u^3 \right]_0^h=\frac{\sqrt{3}hs^2}{12}\)

Using the value we found for $h$, we have:

\(\displaystyle V=\frac{\sqrt{3}\left(s\sqrt{\frac{2}{3}} \right)s^2}{12}=\frac{\sqrt{2}s^3}{12}\)

Now using the given data $s=6$, we obtain:

\(\displaystyle V=\frac{\sqrt{2}6^3}{12}=18\sqrt{2}\)
 

FAQ: Volume of a Tetrahedron: General Slicing Method

What is a tetrahedron?

A tetrahedron is a three-dimensional shape with four triangular faces, six edges, and four vertices.

What is the general slicing method for finding the volume of a tetrahedron?

The general slicing method involves slicing the tetrahedron into smaller pieces that can be easier to calculate and then adding those pieces together to find the total volume.

What are the steps for using the general slicing method to find the volume of a tetrahedron?

The steps for using the general slicing method are:

  • Divide the tetrahedron into smaller pieces, such as triangular prisms or pyramids.
  • Find the volume of each smaller piece using the appropriate formula.
  • Add the volumes of all the smaller pieces together to get the total volume of the tetrahedron.

Can the general slicing method be used for any tetrahedron?

Yes, the general slicing method can be used for any tetrahedron, regardless of its size or shape.

Are there any other methods for finding the volume of a tetrahedron?

Yes, there are other methods such as using the formula V = (1/3) * base area * height or using the determinant of a 3x3 matrix. However, the general slicing method is often the most efficient and straightforward method for finding the volume of a tetrahedron.

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