Editing Tetrahedron (section)

=== Measurement ===
[[File:Tetrahedron.stl|thumb|3D model of a regular tetrahedron]]
Consider a regular tetrahedron with edge length <math>a</math>. Its height is <math display="inline"> {\sqrt{\frac{2}{3}}a} </math>.<ref>Köller, Jürgen, [http://www.mathematische-basteleien.de/tetrahedron.htm "Tetrahedron"], Mathematische Basteleien, 2001</ref> Its surface area is four times the area of an equilateral triangle: <math display="inline"> A = 4 \cdot \left(\frac{\sqrt{3}}{4}a^2\right) = a^2 \sqrt{3} \approx 1.732a^2. </math>{{sfn|Coxeter|1948|loc=Table I(i)}}
The volume is one-third of the base times the height, the general formula for a pyramid;{{sfn|Coxeter|1948|loc=Table I(i)}} this can also be found by dissecting a cube into a tetrahedron and four triangular pyramids.{{sfn|Alsina|Nelsen|2015|p=[https://books.google.com/books?id=FEl2CgAAQBAJ&pg=PA68 68]}}
<math display="block">V = \frac{1}{3} \cdot \left(\frac{\sqrt{3}}{4}a^2\right) \cdot \frac{\sqrt{6}}{3}a = \frac{a^3}{6\sqrt{2}} \approx 0.118a^3.</math>

Its [[dihedral angle]]&mdash;the angle formed by two planes in which adjacent faces lie&mdash;is <math display="inline"> \arccos \left(1/3 \right) = \arctan\left(2\sqrt{2}\right) \approx 70.529^\circ. </math>{{sfn|Coxeter|1948|at=Table I(i)}} {{anchor|Tetrahedral angle}}Its vertex–center–vertex angle—the angle between lines from the tetrahedron center to any two vertices—is <math display="inline"> \arccos \left(-1/3 \right) = 2\arctan\left(\sqrt{2}\right) \approx 109.471^\circ, </math> denoted the '''tetrahedral angle'''.{{sfn|Brittin|1945}} It is the angle between [[Plateau's laws|Plateau borders]] at a vertex. Its value in radians is the length of the circular arc on the unit sphere resulting from centrally projecting one edge of the tetrahedron to the sphere. In chemistry, it is also known as the [[Tetrahedral molecular geometry|tetrahedral bond angle]].

[[File:Вписанный тетраэдр.svg|class=skin-invert-image|thumb|right|upright=1.2|Regular tetrahedron ABCD and its circumscribed sphere]]
The radii of its [[circumsphere]] <math> R </math>, [[insphere]] <math> r </math>, [[midsphere]] <math> r_\mathrm{M} </math>, and [[Exsphere (polyhedra)|exsphere]] <math> r_\mathrm{E} </math> are:{{sfn|Coxeter|1948|loc=Table I(i)}}
<math display="block"> \begin{align}
 R = \sqrt{\frac{3}{8}}a, &\qquad r = \frac{1}{3}R = \frac{a}{\sqrt{24}}, \\
 r_\mathrm{M} = \sqrt{rR} = \frac{a}{\sqrt{8}}, &\qquad r_\mathrm{E} = \frac{a}{\sqrt{6}}.
\end{align}
</math>
For a regular tetrahedron with side length <math> a </math> and circumsphere radius <math> R </math>, the distances <math> d_i </math> from an arbitrary point in 3-space to its four vertices satisfy the equations:{{sfn|Park|2016}}
<math display="block"> \begin{align}\frac{d_1^4 + d_2^4 + d_3^4 + d_4^4}{4} + \frac{16R^4}{9}&= \left(\frac{d_1^2 + d_2^2 + d_3^2 + d_4^2}{4} + \frac{2R^2}{3}\right)^2, \\
4\left(a^4 + d_1^4 + d_2^4 + d_3^4 + d_4^4\right) &= \left(a^2 + d_1^2 + d_2^2 + d_3^2 + d_4^2\right)^2.\end{align}</math>

With respect to the base plane the [[slope]] of a face (2{{sqrt|2}}) is twice that of an edge ({{sqrt|2}}), corresponding to the fact that the ''horizontal'' distance covered from the base to the [[Apex (geometry)|apex]] along an edge is twice that along the [[Median (geometry)|median]] of a face. In other words, if ''C'' is the [[centroid]] of the base, the distance from ''C'' to a vertex of the base is twice that from ''C'' to the midpoint of an edge of the base. This follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other (see [[centroid#Proof that the centroid of a triangle divides each median in the ratio 2:1|proof]]).

Its [[solid angle]] at a vertex subtended by a face is <math display="inline"> \arccos\left(\frac{23}{27}\right) = \frac{\pi}{2} - 3\arcsin\left(\frac{1}{3}\right) = 3\arccos \left(\frac{1}{3}\right)-\pi, </math> or approximately 0.55129 [[steradian]]s, 1809.8 [[square degree]]s, and 0.04387 [[Spat (angular unit)|spats]].