Height of a tetrahedron

Question

How do I calculate the height of a regular tetrahedron having side length $1$ ?

Just to be completely clear, by height I mean if you placed the shape on a table, how high up would the highest point be from the table?

Answer 1

The first thing you need to do is to note that the apex of a regular tetrahedron lies directly above the center of the bottom triangular face. Thus, find the length of the segment connecting the center of an equilateral triangle with unit length to a corner, and use the Pythagorean theorem with the length of an edge as the hypotenuse, and the length you previously derived as one leg. The height you need is the other leg of the implied right triangle.

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Here's a view of the geometry:

and here's a view of the bottom face:

In the second diagram, the face is indicated by dashed lines, and the (isosceles) triangle formed by the center of the triangle and two of the corners is indicated by solid lines.

Knowing that the short sides of the isosceles triangle bisect the 60° angles of the equilateral triangle, we find that the angles of the isosceles triangle are 30°, 30° and 120°.

Using the law of cosines and the knowledge that the longest side of the isosceles triangle has unit length, we have the equation for the length $\ell$ of the short side (the length from the center of the bottom face to the nearest vertex):

$$1=2\ell^2-2\ell^2\cos 120^{\circ}$$

Solving for $\ell$, we find that the length from the center of the bottom face to the nearest vertex is $\frac{1}{\sqrt{3}}$, as indicated here.

From this, the Pythagorean theorem says that the height $h$ (the length from the center of the bottom face) satisfies

$$h^2+\left(\frac{1}{\sqrt{3}}\right)^2=1$$

Solving for $h$ in the above equation, we now find the height to be $\sqrt{\frac23}=\frac{\sqrt{6}}{3}$, as mentioned here.

Answer 2

Consider the tetrahedron inscribed in the unit cube, with vertices at (0,0,0), (1,1,0), (0,1,1), (1,0,1). Its height is the distance from (0,0,0) to the centre of the opposite face, which is given by the equation $x+y+z = 2$. Thus its height is $\frac{2}{\sqrt 3}$, and since the edges of this tetrahedron have length $\sqrt 2$, the height of a regular tetrahedron with side $x$ is $x \sqrt{\frac{2}{3}}$.

Answer 3

You can also use trig based on the dihedral angle between two faces of the tetrahedron.

Writing $ABC$ for the base triangle, $O$ for the apex, $K$ for the center of $ABC$ (the foot of the perpendicular dropped from $O$), and $M$ for the midpoint of (for instance) side $BC$, we have a right triangle $OKM$ with right angle at $K$. So,

$$\text{height of tetrahedron} = |OK| = |OM|\sin{M}$$

$OM$ is the height of the (equilateral) face $OBC$, measuring $\frac{\sqrt{3}}{2}s$, where $s$ is the length of a side.

As for the measure of angle $M$ ... Note that this is the dihedral angle between faces $OBC$ and $ABC$; it is also the angle between (congruent) segments $OM$ and $AM$ in triangle $OMA$. We can use the Law of Cosines as follows:

$$\begin{eqnarray} |OA|^2 &=& |OM|^2 + |AM|^2 - 2 |OM||AM|\cos{M} \\ s^2 &=& \left(\frac{\sqrt{3}}{2}s\right)^2 + \left(\frac{\sqrt{3}}{2}s\right)^2 - 2 \left(\frac{\sqrt{3}}{2}s\right)\left(\frac{\sqrt{3}}{2}s\right) \cos{M} \\ s^2 &=& \frac{3}{4} s^2 + \frac{3}{4}s^2 - 2 \frac{3}{4} s^2 \cos{M} \\ 1 &=& \frac{3}{2} - \frac{3}{2} \cos{M} \\ \frac{-1}{2} &=& - \frac{3}{2} \cos{M} \\ \frac{1}{3} &=& \cos{M} \;\;\; (**)\\ \Rightarrow \sqrt{1-\left(\frac{1}{3}\right)^2} = \frac{\sqrt{8}}{3} =\frac{2\sqrt{2}}{3}&=& \sin{M} \end{eqnarray}$$

Therefore,

$$\text{height of tetrahedron} = |OK| = |OM|\sin{M} = \frac{\sqrt{3}}{2} s \cdot \frac{2\sqrt{2}}{3} = \frac{\sqrt{6}}{3}s$$

(**) This cosine is the reason I posted this approach. It's sometimes handy to know (as in this problem); even better, it's easy to remember, because it turns out that it fits a simple pattern (which might be more-likely to impress interviewers):

$$\begin{eqnarray} \cos\left({\text{angle between two sides of a regular triangle}}\right) &=& \frac{1}{2}\\ \cos\left({\text{angle between two faces of a regular tetrahedron}}\right) &=& \frac{1}{3}\\ \cos\left({\text{angle between two facets of a regular n-simplex}}\right) &=& \frac{1}{n} \end{eqnarray}$$

(Who would've suspected, upon first encountering it, that the "$2$" in "$\cos{60^{\circ}}=\frac{1}{2}$" was actually a reference to the dimension of the triangle?)

Answer 4

The centroid, a vertex, and the midpoint of the base form a right triangle on the base of the tetrahedron. Let the distance from a vertex to the centroid be $d$, and the side length be $2$. Observing that the side lengths in this $30º-60º-90º$ triangle are $1 - \sqrt{3} - 2$, $d^2 = 1^2 + \left(\frac{\sqrt3}{3}\right)^2 = \frac{4}{3}$ by Pythagoras: the centroid is one-third the distance from the base to the vertex.

Then using the other right-angled triangle, the height $h^2 = 2^2 - d^2$, where the edges are all $2$, so $h = \sqrt{4 - 4/3} = \sqrt{8/3}$. Now the height of the original tetrahedron with side length $1$ is half of $\sqrt{8/3}$ by similarity, which is $\frac{2\sqrt{2/3}}{2} = \sqrt{2/3}$.

Answer 5

Here is an answer without words following the lines of J.M.'s answer:

Answer 6

My solution is definitely not a fast way, bit it does allow to solve it mentally (without any paper/pen/computer) and without cosines rule, and that was my challenge.

• The tetrahedron fits in a cube. Actually 2 tetrahedra fit in the same cube, which forms a stellated octahedron (sometimes called star tetrahedron).

• This is a composed figure which consists of 1 octahedron and 8 tetrahedra, all the edges in the figure have the same length.

• The edges of the stellated octahedron are the half of the sides of the original tetrahedron. So when the original tetrahedron edges are 2, all edges of the internal octahedron, as well as the smaller tetrahedra, are 1 (see image).

• Now we have to find the height of one of those smaller tetrahedra. As the inner octahedron consists of squares with sides 1, the height of the octahedron (and therefore also the encompassing cube) according to the pythagorean theorem, is $\sqrt{2}$.

• Now we can calculate the total height of the stellated octahedron (from point to point), because it is the diagonal of the encompassing cube, of which we already know the sides ($\sqrt{2}$ and $2$). Using the pythagorean theorem we get $\sqrt{ 2^2 + (\sqrt{2})^2} = \sqrt{4 + 2} = \sqrt{6}$

• When you hold the stellated octahedron between the two outer points you see 3 layers of tetrahedra. We only have to divide the height ($\sqrt{6}$) by $3$ to get the height of a tetrahedron with side 1: $\sqrt{6}/3 = 0.81649658092$ (which is the same as $\sqrt{2/3}$)

Here is a visual explanation