The Regular Tetrahedron

Regular tetrahedron is one of the regular polyhedrons. It is a triangular pyramid whose faces are all equilateral triangles.

Properties of a Regular Tetrahedron

There are four faces of regular tetrahedron, all of which are equilateral triangles.
There are a total of 6 edges in regular tetrahedron, all of which are equal in length.
There are four vertices of regular tetrahedron, 3 faces meets at any one vertex.
A regular tetrahedron can circumscribe a sphere that is tangent to all the faces of the tetrahedron.
A regular tetrahedron can be inscribed in a sphere that passes through all the vertices of tetrahedron.
The center of the inscribed sphere, the center of the circumscribing sphere, and the center of the regular tetrahedron itself are coincidence.

Formulas for Regular Tetrahedron

Area of one face, A_b

$A_b = \frac{1}{2}a^2 \sin \theta$

$A_b = \frac{1}{2}a^2 ( \frac{\sqrt{3}}{2} )$

$A_b = \dfrac{a^2 \sqrt{3}}{4}$

Total area, A_T
$A_T = 4A_b$

$A_T = 4 ( a^2 \sqrt{3} / 4 )$

$A_T = a^2 \sqrt{3}$

Slant height, L
$L = a \sin 60^\circ$

$L = a ( \frac{\sqrt{3}}{2} )$

$L = \dfrac{a\sqrt{3}}{2}$

Altitude, h
$(\frac{2}{3}L)^2 + h^2 = a^2$

$\frac{4}{9}L^2 + h^2 = a^2$

$\frac{4}{9}(a \sqrt{3} / 2)^2 + h^2 = a^2$

$\frac{4}{9}( \frac{3}{4} a^2 ) + h^2 = a^2$

$\frac{1}{3}a^2 ) + h^2 = a^2$

$h^2 = \frac{2}{3}a^2 )$

$h = a \sqrt{\frac{2}{3}}$

$h = a \sqrt{\frac{2}{3} \times \frac{3}{3}}$

$h = a \sqrt{\frac{6}{9}}$

$h = \dfrac{a\sqrt{6}}{3}$

Volume, V
$V = \frac{1}{3}A_b \, h$

$V = \dfrac{1}{3} \left( \dfrac{a^2 \sqrt{3}}{4} \right) \left( \dfrac{a\sqrt{6}}{3} \right)$

$V = \dfrac{a^3 \sqrt{18}}{36}$

$V = \dfrac{a^3 \sqrt{9(2)}}{3(12)}$

$V = \dfrac{a^3 (3 \sqrt{2})}{3(12)}$

$V = \dfrac{a^3 \sqrt{2}}{12}$

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