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Regular Tetrahedron 3-D Formula

Regular tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular or equilateral and is one of the Platonic solids.

The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle, so a tetrahedron is also known as a triangular pyramid. Any of the four faces can be considered the base.

The tetrahedron is the only convex polyhedron that has four faces. For any tetrahedron there exists a sphere or circumsphere such that the tetrahedron's vertices lie on the sphere. Formulas for regular tetrahedron include the following,

For a regular tetrahedron of edge length a:

Base plane area          A_0={\sqrt{3}\over4}a^2 \,

Surface area   A=4\,A_0={\sqrt{3}}a^2 \,

Height h={\sqrt{2\over3}}a \,

Volume            V={1\over3} A_0h ={\sqrt{2}\over12}a^3 \,

Angle between an edge and a face    \arccos\left({1 \over \sqrt{3}}\right) = \arctan(\sqrt{2}) \,
(approx. 54.7356°)

Angle between two faces       \arccos\left({1 \over 3}\right) = \arctan(2\sqrt{2}) \,
(approx. 70.5288°)

Angle between the segments joining the center and the vertices    \arccos\left ({-1\over3}\right )\,
(approx. 109.4712°)

Solid angle at a vertex subtended by a face   3 \arccos\left ({1\over3}\right ) - \pi \,
(approx. 0.55129 steradians)

Radius of circumsphere          R=\sqrt{{3\over8}}\,a \,

Radius of insphere that is tangent to faces    r={1\over3}R={a\over\sqrt{24}} \,

Radius of midsphere that is tangent to edges            r_M=\sqrt{rR}={a\over\sqrt{8}} \,

Radius of exspheres    r_E={a\over\sqrt{6}} \,

Distance to exsphere center from a vertex   \sqrt{{3\over2}}\,a \,

What is a Tetrahedron?

A tetrahedron is a three-dimensional figure with four equilateral triangles. If you lift up three triangles (1), you get the tetrahedron in top view (2). Generally it is shown in perspective (3).

If you look at the word tetrahedron (tetrahedron means "with four planes"), you could call every pyramid with a triangle as the base a tetrahedron.

However the tetrahedron is the straight, regular triangle pyramid on this website.
 
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