Formulas For A Regular Tetrahedron
The following Cartesian coordinates define the four vertices of a tetrahedron with edge-length 2, centered at the origin:
- (±1, 0, -1/√2)
- (0, ±1, 1/√2)
For a regular tetrahedron of edge length a:
| Base plane area | |
| Surface area | |
| Height | |
| Volume | |
| Angle between an edge and a face | (approx. 54.7356°) |
| Angle between two faces | (approx. 70.5288°) |
| Angle between the segments joining the center and the vertices, also known as the "tetrahedral angle" | (approx. 109.4712°) |
| Solid angle at a vertex subtended by a face | (approx. 0.55129 steradians) |
| Radius of circumsphere | |
| Radius of insphere that is tangent to faces | |
| Radius of midsphere that is tangent to edges | |
| Radius of exspheres | |
| Distance to exsphere center from a vertex |
Note that with respect to the base plane the slope of a face is twice that of an edge, corresponding to the fact that the horizontal distance covered from the base to the apex along an edge is twice that along the 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 proof).
Read more about this topic: Tetrahedron
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