Tetrahedron properties

For a regular tetrahedron of edge length a:

 Face area

Surface area^[2]

Height of pyramid^[3]

Edge to opposite edge distance

Volume^[2]

Face-vertex-edge angle

Face-edge-face angle^[2]
Edge central angle,^[4][5] known as thetetrahedral angle
Solid angle at a vertex subtended by a face
Radius of circumsphere^[2]

Radius of insphere that is tangent to faces^[2]

Radius of midsphere that is tangent to edges^[2]

Radius of exspheres

Distance to exsphere center from the opposite vertex

Find the volume of an irregular tetrahedron form its edges:

Suppose you are given the 6 sides of an irregular tetrahedron and you need to find the volume consumed by it.

Let the given sides to be u, v, w, W, V, U. Here, (u, U), (v, V), (w, W) are considered to be opposite edge pairs ( opposite edges means the edges which do not share common vertices ). Now the volume can be found from the following formula:

Let:
u′ = v² + w² - U²
v′ = w² + u² - V²
w′ = u² + v² - W²
Now:
volume = ¹⁄₁₂ × √(4u²v²w² - u²u′² - v²v′² - w²w′² + u′v′w′)

This formula is derived from the determinant which can be found here for more reading. As the formula is symmetric, the ordering of the pairs won't make any change to the formula.