Tetrahedron Volume from 4 Vertices

Find the volume of a tetrahedron from the coordinates of its four vertices using the scalar triple product, plus each face area, centroid, surface area and total edge length.

Input

Enter the coordinates of four points in 3D space to compute the tetrahedron volume with the scalar triple product, along with each face area, the centroid and the surface area.

Point P1

Point P2

Point P3

Point P4

Result

Volume

0.166667

Surface area

2.366025

Total edge length

7.242641

Centroid

(0.25, 0.25, 0.25)

Face areas

Face P1P2P3

0.5

Face P1P2P4

0.5

Face P1P3P4

0.5

Face P2P3P4

0.866025

The volume is computed with the scalar triple product, the absolute value of (P2 minus P1) dot ((P3 minus P1) cross (P4 minus P1)) divided by 6.

How it works

The volume comes from the scalar triple product. Using P1 as the base point, build three vectors a=P2 minus P1, b=P3 minus P1 and c=P4 minus P1, then volume equals the absolute value of a dot (b cross c) divided by 6.
b cross c is the cross product. Its dot product with a gives the signed volume of the parallelepiped spanned by the three vectors, and one sixth of its magnitude is the tetrahedron volume.
When the volume is near zero the four points are coplanar and do not form a tetrahedron. A note is shown in that case.
Each face area is half the magnitude of the cross product of two edge vectors of that triangular face. The surface area is the sum of the four face areas.
The centroid is the arithmetic mean of the four vertex coordinates. The edge total is the sum of the lengths of all six edges.
Coordinate units are arbitrary. Volume is in cubic units, area in square units and length in the same unit as the inputs.

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