The tetrahedron is a special situation of the pyramid in the feeling that it is a triangular pyramid (i.e. all of the faces are triangles, consisting of the base polygon). The general qualities of pyramids are dealt with in the web page entitled "Pyramids". Top top this page, we will be looking specifically at the properties of tetrahedra. Choose all pyramids, the tetrahedron is a polyhedron (i.e. A three-dimensional geometrical form with flat faces and also straight edges). It has four (4) encounters (the indigenous tetra has its origins in the Greek language, and way four), six (6) edges, and also four (4) vertices. What distinguish the tetrahedron from other pyramids is that all that its encounters are triangles. Together you have the right to see native the illustration below, the tetrahedron has a triangular base (any one of the tetrahedron"s four faces can it is in designated to it is in the base), and also three triangular sides that affix the base to the apex. Every vertex of the tetrahedron is mutual by three of its triangle faces.




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all of the deals with of a tetrahedron are triangles, consisting of the base


The tetrahedron is the only polyhedron that has four faces. That is also the only straightforward polyhedron that has no polyhedron diagonals (i.e. No challenge diagonals or room diagonals). One isosceles tetrahedron is a special instance of the basic tetrahedron for which all four of the triangular faces are congruent. A regular tetrahedron is a special situation of both the basic tetrahedron and the isosceles tetrahedron for which all four triangular encounters are not just congruent, yet are additionally equilateral triangles, i.e. Triangles because that which all three sides are the exact same length, and also all internal angles space sixty degrees (60°). The illustration below shows an isosceles tetrahedron.


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The encounters of one isosceles tetrahedron space congruent triangles


In one isosceles tetrahedron, each pair of the opposite edges space congruent (i.e. Equal in length). With recommendation to the above illustration, therefore, we have:

a=a" b=b" c=c"

native this, it must be relatively easy to check out that the four faces of the isosceles tetrahedron space congruent, because each the the four triangular faces is written of among the following combinations the sides:

a, b, c a, b", c" a", b", c a", b, c"

If you look very closely at the illustration, you should be able to see that the three confront angles that satisfy at every vertex of the tetrahedron consists the angle between sides a (or a") and also b (or b"), add to the angle in between sides a (or a") and c (or c"), to add the angle between sides b (or b") and c (or c"). Because each peak is thus efficiently formed by three different internal angles from triangle that room congruent, and also since the three interior angles of a triangle always add up come one hundred and also eighty degrees (180°), then the challenge angles at every vertex of one isosceles tetrahedron must also sum to one hundred and also eighty degrees.

together we have currently mentioned, the regular tetrahedron is a special case of the isosceles tetrahedron for which every four faces are congruent equilateral triangles. The regular tetrahedron is just one of the 5 platonic solids (a platonic solid is a constant convex polyhedron v all encounters being regular congruent polygons, and the same variety of faces meeting at each vertex). The constant tetrahedron is the only regular polyhedron with no parallel faces, and also has a number of other attributes that follow from the fact that every one of its encounters are congruent equilateral triangles:

every edges space of the same length all inner angles have actually a size of 60° the 4 vertices space equidistant indigenous one one more opposite edges are perpendicular (at right-angles) come one an additional four axes of the opposite pass with each vertex and the centroid the its opposite challenge three axes of symmetry pass with the midpoints of opposite edge the dihedral edge between any two encounters has a magnitude of (circa) 70.53° the angle in between an edge and also a confront has a size of (circa) 54.74°

A constant tetrahedron have the right to be formed using six of the confront diagonals the a cube, as displayed below. The vertices that the tetrahedron are coincident with 4 of the cube"s vertices. The figure plainly illustrates some of the qualities of a continuous tetrahedron detailed above. For example, since each leaf of the tetrahedron is also one of the cube"s confront diagonals, the edge of the tetrahedron have to be same in length. Furthermore, due to the fact that each that the tetrahedron"s vertices is connected to every other vertex by one edge the is additionally one that the cube"s challenge diagonals, they have to be equidistant. Finally, every pair of opposite edges for the tetrahedron is composed of a pair of alternating face diagonals belonging come opposite encounters of the cube. Lock must therefore be perpendicular come one another.


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A continuous tetrahedron shares 4 vertices with a cube


Since all of the encounters of a continuous tetrahedron space equilateral triangles, all of the interior angles of the tetrahedron will certainly be sixty degrees (60°), and also the amount of the challenge angles because that the three encounters meeting at any type of vertex will certainly be one hundred and also eighty degrees (180°). The axes that symmetry room of specific significance when we room dealing with continual tetrahedra. Remember that there is an axis the symmetry that connects every vertex of the regular tetrahedron with the centroid the its opposite confront (see the illustration below). By definition, the heat passing with any crest of a tetrahedron and the centroid that the challenge opposite that vertex is dubbed a median. A tetrahedron thus has four (4) medians. Because that a regular tetrahedron, each mean is coincident with one of its axes of symmetry.


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because that a continuous tetrahedron, each average is coincident with an axis of the opposite


A similar situation occurs once we take into consideration the axis of symmetry the connects the mid-point of 2 opposite edges of a consistent tetrahedron (see the illustration below). By definition, the line passing through the mid-points of 2 opposite edges for any tetrahedron is referred to as a bimedian. A tetrahedron will therefore have three (3) bimedians, due to the fact that it has three bag of the contrary edges. Again, for a regular tetrahedron, each bimedian is coincident with one of its axes of symmetry.


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for a continual tetrahedron, each bimedian is coincident v an axis of symmetry


The 4 medians and also three bimedians the a tetrahedron (and as such by definition the seven axes the symmetry because that a consistent tetrahedron) all crossing at a single point dubbed the centroid (or geometric centre) that the tetrahedron. If the tetrahedron were a solid object of uniform density, the centroid would certainly be that object"s center of mass.

we mentioned above that a consistent tetrahedron might share 4 vertices v a cube, however there is another (non-regular) form of tetrahedron dubbed a trirectangular tetrahedron because that which all three confront angles at one of its vertices are right-angles. You have the right to visualize what this tetrahedron could look prefer by imagining what friend would acquire if you sliced turn off one corner of a cube. The illustration below attempts to clarify this idea.


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A trirectangular tetrahedron has a solitary vertex in ~ which every three confront angles room right-angles


For all tetrahedra, there exists a sphere dubbed the circumsphere which completely encloses the tetrahedron. The tetrahedron"s vertices every lie top top the surface of that circumsphere. The suggest at the center of the circumsphere is dubbed the circumcentre.


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all of the tetrahedron"s vertices lied on the surface of that circumsphere


keep in mind that for a regular tetrahedron through an edge size of a, the radius of the circumsphere is provided by the complying with formula:

Radius that circumsphere=√3a
8

There additionally exists for all tetrahedra a sphere well-known as the insphere that is totally enclosed through the tetrahedron. The insphere is tangent to each of the tetrahedron"s 4 faces, i.e. The insphere touch the surface of each challenge at a single point. The allude at the centre of the insphere is dubbed the incentre.




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note that because that a continuous tetrahedron with an edge size of a, the radius of the insphere (which is precisely one third of the radius of the circumsphere) is given by the following formula:

Radius that insphere=a
√24

keep in mind that for any isosceles tetrahedron (of i beg your pardon the continual tetrahedron is a unique case) the circumsphere and insphere are concentric (i.e. Castle share a common centre), which means that the circumcentre and also incentre will be coincident.

obtaining down to more mundane issues, the volume and also surface area of a tetrahedron deserve to be uncovered using the same techniques as for any other pyramid. The volume V the a tetrahedron is calculated as one third of its basic area B (regardless of which of the four encounters is taken to be the base) multiplied by its height h. We deserve to formalise this relationship as:

V=1Bh
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note that because that a regular tetrahedron through an edge size of a, the volume deserve to be discovered using the following alternate formula:

V=√2a3
12

recognize the surface ar area of a (non-regular) tetrahedron is typically a little much more complicated, due to the fact that the four triangular encounters may all have different areas. If this is the case, the area the each confront must be discovered separately, and the values included together to obtain the full surface area. If the tetrahedron is isosceles, life is somewhat simpler due to the fact that the four encounters will be congruent, and will therefore all have the same area. Note that for a regular tetrahedron through an edge size of a, the total surface area A deserve to be discovered using the complying with formula:

A=√3a2

The tetrahedron is of attention to mathematicians for its own sake, many thanks to its reasonably unique characteristics amongst polyhedra. That is likewise studied by scientists, engineers, and also practitioners in other locations for a variety of reasons. The is of attention in the field of chemistry, because that example, due to the fact that the molecules of miscellaneous chemical link (water and methane, for example) have a tetrahedral structure. In the field of electronics, the tetrahedron is the significance since silicon crystals have actually a tetrahedral structure. This is due to the truth that silicon atoms have a valence of four (the valence of one atom shows the number of chemical bonds the can form with other atoms). Silicon is the semiconductor product most typically used in the manufacturing of electronic circuit boards and also components, so its crystalline structure has actually naturally to be the subject of considerable study.