What is a Tetrahedron shape? Everything You Need to Know

Tetrahedron is a word that sounds very fancy and maybe you know the basic shape is a triangular pyramid. But what is a tetrahedron shape and is it just as fancy as the word? I found all the most interesting features of tetrahedra (which is the plural of tetrahedron), without too much cumbersome jargon so you can enjoy much of what the tetrahedron shape has to offer.

There are other places where you can read even more in depth, but it doesn’t necessarily mean you’ll learn much. Math jargon and laundry lists of specific features few understand. I’ve created a story that explains to you full what is a tetrahedron shape, for the curious knowledge-seeker.

What is a Tetrahedron shape? Basic Stats

A tetrahedron is a triangular pyramid, in general, an enclosed solid with four triangles as the sides. The most popular tetrahedron shape is one made of four equilateral triangles. See the variety of forms a tetrahedron can take, but they all do share many characteristics.

Basic Stats of a Tetrahedron

The GENERAL tetrahedron is just four triangles, making an enclosed shape. They don’t have to be the same size, or have anything else in common. No matter what, a tetrahedron will have six edges, four faces, and four vertices. Let’s narrow the focus to a regular tetrahedron, with all four faces the same, equilateral triangle (a 60°-60°-60° triangle with all sides of equal length.) Other species we’ll look at towards the end, so hold your horses.

Here you can move around and manipulate a disphenoid, where the faces are all identical but not equilateral.

Not like the Egyptian pyramids, which were square based. Greeks used the word “tetragon”, and Euclid simply calls the tetrahedron he describes as a pyramid in his 300 B.C. Elements.

There are so many ways to classify polyhedra. I found Maeder index, Wenniger index, Coxeter indes, Har’El index, Schlå¨å¨fli symbol, and Wythoff symbol. What is this, an RPG? It’s not necessary to understand its classifications to see the symmetries and main properties of the tetrahedron.

The tetrahedron has the numerical archetype of Fourness

For millennia now, scholars and polymaths have pondered the tetrahedra. Like the other regular solids, the tetrahedra is foundational to some sort of understanding all more complex construction. The tetrahedron associates with the element of fire. In Kepler’s 1597 Mysterium Cosmographium, the tetrahedron is inside a sphere describing the orbit of Jupiter, and is situated around the sphere describing Mars. The tetrahedron in many ways represents the simplest of the regular solids. In this way many believe it to blend the elements of fire and Earth, representing creation (Earth) through purification (fire), a movement from nothing to something.

Here is more about the numerical archetype of “four,” which in general is a grounded than fiery vibe (more like three).

~~> Fourness Archetype Description + Examples

The tetrahedron encompasses creation of foundation through ignition. And we’ll see more connections with its spark in the tetrahedral molecules examples.

Tetrahedron Platonic Solid Status

To be an official Platonic Solid™ (of which there are only five), a solid must be:

• made of regular, convex polygons
• where at each vertex (corner) the same number of the polygon meet

To be regular means all the sides of each face are the same length. To be convex the inside angles are all greater than ninety degrees. You can tell if a solid is not convex because it would have a corner pointing INTO it, for example, a spot where water could collect if you held it a certain way. In contrast, you can in no way hold the tetrahedron and collect any water, it will all run off.

At each corner of the tetrahedron is the meeting of three triangles. Again, each of those triangles are regular, also called equilateral triangles, with the same length for all three sides. When a triangle is equilateral, we already know everything about the ratio of its sides just from Pythagorean theorem.

Surface area, volume, and other metrics of a Tetrahedron

For the tetrahedron’s vital stats, they can all be expressed as proportional with one of the edge lengths.

The Two Possible Nets for a Regular Tetrahedron

A regular tetrahedron has two nets that are equivalent when folded up.

Sometimes a dihedral angle is mentioned when studying solids. This just refers to an angle between two planes. A tetrahedron’s dihedral angle is about 70.52°.

A unique feature of tetrahedral is that they are self-dual. The dual of a tetrahedron is another, smaller tetrahedron.

Of the other Platonic solids, the cube and octahedral are duals of each other, and the icosahedron and dodecahedron are duals of each other. All pyramids no matter the shape of the base are self dual. You may have also heard of a star tetrahedron, I revisit this at the end.

Symmetries of a Tetrahedron (Intro to Point Groups)

This is all you need to know: Tetrahedra make up the symmetry point group T_d. Okay, so what does that mean?

If you get into the mathematical definitions of symmetry, it starts to seem like a foreign language, unless you’re versed in the areas of math called group theory. (Group theory just aims to take really abstract mathematical objects and “group” them according to their symmetric properties.) Geometric symmetry is super relevant for crystal structure, since “crystal systems” (the geology/physics way of categorizing symmetry) and “point groups” (the math way of categorizing symmetry) are exactly the same. As we build up this knowledge we started with generalizing symmetry and going into rotational symmetry in some of my prior posts

~~> Different types of lines of symmetry
~~> How to see Rotational symmetry
~~> Examples of rotational symmetry in art

A point group has a list of transformations that a shape in the group can undergo and look the same. For example, if you take a plain piece of paper and flip it over, it will look the same as before you flipped it. So the transformations enumerate the different ways you can turn or flip a structure and go unnoticed.

In a tetrahedron, all the vertices are an equal distance from one another and completely identical. If you are standing at any vertex of a tetrahedron you wouldn’t be able to tell if you teleported to another one. Any transformation that takes one vertex and maps it to another will work for the regular tetrahedron. This is like setting a tetrahedron on the table, then moving it so any of the other three vertices take the place of another. You wouldn’t be able to tell the difference. When you see this, it’s easier to understand why all the symmetries add up to 24.

The 24 Symmetries of the Tetrahedron

Remember of the point groups, which describe all the inner shapes of crystals with which electromagnetic energy interacts. The tetrahedral point group (T_d) is the simplest of 32 point groups each defining a set of symmetry operations. So if you take your time with understanding the symmetries of the tetrahedron, you’ll be better positioned if you would like to internalize more of the possible symmetries in molecules or crystals later on.

The 24 T_d symmetries are:

• E (identity, if you do nothing it’s the same)
• 8 C₃ (axes where you can rotate 120°)
• 3 C₂ (axes where you can rotate 180°)
• 6 S₄ (“improper rotation” – rotation of 90° then reflect perpendicularly)
• 6 σ_d (reflecting across “dihedral mirror planes”)

It’s super useful to actually pretend the tetrahedron is in a cube when thinking about the symmetries. this is just because for most people, imagining a cube and manipulating it is way easier. You can make a model of a tetrahedron, find a cube shaped object, or simply try to doodle in 3D, to better internalize all the symmetries of the tetrahedron.

Rotational Symmetry of a Tetrahedron

Rotational symmetry angles tell you how much you can TURN the object by to achieve identical-ness. C₂ means you can turn the object twice in a 360°. Every 180° the pattern repeats. C₃ means you can turn the object three times in every 360°, or 120°. (360°/n = the rotational symmetry angle around the axis called C_n)

The 8 C₃ are on the body diagonals of a cube enclosing the tetrahedron. Imagine sticking a wire from one corner to the one diagonally through the bulk of the cube. Those C₃ axes go through each of the tetrahedron’s four vertices to the opposite face. To see the C₃ axes, you can also imagine the tetrahedron is in the “tripod position,” with one face flat on the surface, three of its four corners on the table, and one sticking up, looking like a classic triangular pyramid. We can rotate the base of the tripod.

A tetrahedron’s C₂ axis goes through each midpoint of an edge, to the edge opposite (three in total, one for each two of the six edges). In the cube, these are through the 3 face centers, where a 180° turn will give the identical figure. Since a 180° turn is identical whether you go clockwise or counterclockwise, these symmetries are not doubled like the C₃ are and only count for three.

So for rotational symmetry, the tetrahedron has seven axes of rotational symmetry, three 180° turns, and four 120° turns, each of which can go clockwise or counterclockwise, for a total of 11 ways to make rotational symmetry in a regular tetrahedron. Here is a fun applet where you can play with the rotational symmetries of any Platonic solid.

Reflection Symmetry of a Tetrahedron

The σ_d symmetries are mirror reflections across dihedral planes. Dihedral just means the mirror plane lies between two of the plane faces. Of the four vertices, simply pick two at a time (there are six ways to do this). The mirror plane cuts right in the middle of the space between two faces of the tetrahedron. These planes also go through each edge of the tetrahedron, bisecting the six face diagonals of the cube. We can flip the tetrahedron across these planes to identify an identical image.

Note that the σ_d planes cut through each edge, and on the other side they end up cutting the midpoint of the opposite edge. You can especially see the mirror planes in this applet (select T_d, σ_d, check the box to show the axis, then rotate the tetrahedron with your mouse.)

The S₄ symmetry is known as an improper axis, which consists of a rotation and then a reflection. The rotation axes for these transformations are the same three C₂ axes, rotated only 90° this time. Then, perform a reflection across the plane perpendicular to the axis.

There are six S₄ symmetries because there are two for each of the three C₂ axes: one for a 90° clockwise rotation, and the other for the counterclockwise rotation.

You can also look and play with the all tetrahedral symmetries in this applet by selecting T_d from the list.

There are even more tetrahedral symmetries than just T_d. These are more specific subsets within tetrahedral symmetry, like: chiral tetrahedral T, achiral tetrahedral T_d (the specific term for what we just covered), and pyrectohedral T_n.

Appearances in Chemistry, Math, and Earth Science

What about the tetrahedron’s significance in the “real world”? The tetrahedron shape on the molecular scale translates to the shared properties of tetrahedral forms in nature. In biology, the main example of the tetrahedron shape is in certain radiolaria, which are micro-organisms that make geometric mineral skeletons.

Silicon-Oxygen Tetrahedron, the molecular template for all Silicates

A neutral atom has a certain number of electrons on its outer cloud that determines how it will bond and form compounds. Silicon has a valence of four, and oxygen has a valence of six. Silicon and oxygen forma neutral compound silicon dioxide, SiO₂, which is also the main component of glass and quartz. Silicon and oxygen can also combine to make a tetrahedral charged ion SiO₄^4- . (Learn about silicate minerals in general which all have this ion.) This charged silicon-oxygen ion is often found in nature bonded to make neutral rocks, like in amphiboles, clays, mica, and other silicates abundant on Earth.

Some people might say water is tetrahedral and they are wrong. The bond angle in water is 104.5°, and as we’ll see below, tetrahedral molecules have a perfectly 109.5° bond angle. These tetrahedral molecules do tend to be non-polar like water though, meaning their charge is evenly distributed across them.

Methane, the perfect tetrahedral molecule

Silicon-oxygen tetrahedrons are usually bonded to a bunch of other stuff in a rock matrix. The most prevalent tetrahedral molecule on its own is methane, with four hydrogen atoms all bonded to a carbon atom. Each hydrogen molecule wants to get as far away from the others as possible, so they evenly space out around in three dimensions. This applies for any AB₄ end-members. In two dimensions the even spacing of 4 points would be 90°. But in three dimensions we have a little more room.

Methane, CH₄, has a neutral charge and is by far the best example of a perfect 109.4° bond angle in a naturally occurring molecule. Many others come close though. Halomethanes are when the four hydrogen atoms are replaced with a halogen. This is the second-to-last column in the periodic table, such as fluorine, chlorine, and bromine. These halomethanes can make propellants and refrigerants and all have tetrahedral molecular structure. There are even some marine organisms that make halomethanes as a byproduct of metabolism. The halomethanes can interact with ozone and include the CFCs if you’ve ever heard of those. It’s the stuff in hairspray, for example, when people talk about it not being environmentally safe.

Other Tetrahedral Molecules

Methane CH4	109.5°	gas hydrocarbon power source
Sulfate SO42-	109.5°	surfactent used in food and cosmetics
Permanganate MnO4–	109.5°	oxidizing agent for disinfection
Ammonium NH4+	109.5°	fertilizer, refrigerant, and manufacturing
Perchlorate ClO4–	109.5°	oxidizer for rocket fuel, flares
Silicon-Oxygen Ion SiO44-	109.5°	naturally occurs in rocks with a wide variety of uses
Tetraphosphorus T4	109.5°	formerly a pigment, now for explosives
Titanium Tetrachloride TiCl4	109.5°	used in isolating titanium metal
Phosphate PO43-	*109.4°	fertilizers, food and cosmetics

Some other tetrahedral molecules include TiBr₄, Titanium tetrabromide, a corrosive; TiI₄, Titanium tetraiodide, very volatile and corrosive (intermediate in purifying titanium); CI₄, Carbon tetraiodide, toxic, used in chemical reactions; CCl₄, Carbon tetrachloride, toxic carcinogen and depletes ozone.

Carbon tetrachloride has an entertaining history. It can reduce to chloroform, and used to be used in fire extinguishers, coolants, and many cleaning products. It was also a key ingredient in lava lamps, adding weight to parts of the fluid. There is still carbon tetrachloride, generally inert, allowed in cleaning products. In the 1800s, an experimental anesthetic. It had effects similar to chloroform or ether, a medicine used at that time. Until the 1950s, dry cleaners also used carbon tetrachloride. Then it was used in chemistry experiments as a solvent, but nearly all its uses superseded by discoveries of less toxic alternatives. I found carbon tetrachloride a very interesting archaic molecule that people tried to use for everything, but at the end of the day, too damaging to live on as useful.

Plenty of people died using it in tasks such as developing photos, finishing floors, stripping paint, and so on. It was used as dry shampoo until the 1930s, and people sometimes passed out and even died from the fumes.

You would think this would be enough, even though all tetrahedral molecules but methane above are ions and thus reactive. But there’s more, some specially engineered to be tetrahedral for certain properties. A theme is that these structures are not exactly stable. But there are also tetrahedral structures in solid state crystals as well, where the presence of other molecules stabilizes the energy.

Deriving the 109.5° bond angle in tetrahedral molecules

Since the 109.5° angle can seem a bit random, I can show you where that comes from. All you need to understand this is the distance formula or pythagorean theorem. (For the length of a diagonal line connecting two points, square root the sum of the horizontal and vertical distances, both squared.)

We put a cube in a grid and label all the points with coordinates. Then we can get the length connecting two corners and get the angle from that. We can make a tetrahedron within the cube by simply picking four corners that meet the basic criteria of being equal distance from one another. You can tell because each of these four points is just a face diagonal away from ALL the other three. (Doodling helps too!) We can also confirm that the four points are equidistant using Pythagorean theorem. The face and body diagonal of a cube is √2.

Now we want to find the angle (in 3-D) that any pair of those tetrahedron corner points makes with the center making the “elbow”. In the AB₄ molecules, the A atom would be at the center, and the four B’s at the corners of a tetrahedron. To find the angle we can write the connection between the points with the center as vectors (arrows) and use the “dot product”. The angle is the inverse cosine of the dot product divided by the product of their lengths. We find expressions mathematically for the vectors, their lengths, the dot product, and then feed that in to find the angle.

More types and features of tetrahedra

So far specifically we only covered where all the faces are equilateral triangles. But if we change the shape of the triangle we do get some different features. The faces, edges, and vertices stay the same in number, but the volume and surface area are a little different, depending on what specific triangles makeup the faces.

There’s more than One Kind of Tetrahedron Shape

Can tetrahedra fill space with no gaps?

Regular tetrahedra can not fill space without gaps. One neat regime of tetrahedra is splitting a cube into six tetrahedra, in which there are 2twodifferent orientations of tetrahedra within. Cubes can tile to fill space without gaps, so those two types of tetrahedra (called the 3-orthoscheme) can fill space in what’s called the disphenoid tetrahedral honeycomb.

What is a “star tetrahedron”?

It’s equivalent to a “stellated octahedron.” Stellated means you take the octahedron and put points on each of its faces.

Lots of people in the sacred geometry community love this shape for its symbolism. It has properties of the numerical archetypes of six-ness and three-ness. Right away when looking at the star tetrahedron, you can notice threes and sixes. It is the combination of the two tetrahedron, to make the third, distinct shape, giving it features of duality and triple-polarity. In this context, the star tetrahedron is also called the merkaba. Merkaba comes from ancient Egyptian: mer, meaning light, two counter-rotating fields; ka, meaning spirit; and ba, meaning the physical body. In Hebrew, merkaba means “chariot.” The enclosing cube when laid out with thirteen circles forms Metatron’s cube, associated with teleportation, or a vehicle connecting heaven and Earth.

Here’s more on the archetypes of three and six:

~~> Numerical Archetype of Three
~~> Examples of Numerical Archetype Three
~~> Numerical Archetype Six
~~> Examples of Numerical Archetype Six

You’re Sorry You Asked – What is a tetrahedron shape?

I hope I covered the basics here! All these words and pictures are still only a start of your own experience with this first platonic solid. Anything else you can get from doodling and observing. Observe the properties in the world around you, play around with the math and doodling!

References/sources

Guide to Euclid’s tetrahedra
Pictures of Euclid’s Elements from Brown.edu

Group Theory for Symmetry:
Drago, Russell S. “Physical methods in chemistry.” (1977).
Chemistry LibreTexts on Point Groups

Carbon Tetrachloride: Odabasi M. (2008). “Halogenated Volatile Organic Compounds from the Use of Chlorine-Bleach-Containing Household Products”. Environmental Science & Technology. 42 (5): 1445–51. Bibcode:2008EnST…42.1445O. doi:10.1021/es702355u. PMID 18441786.