Tetrahedron

Builds on Convex hull Plane Euclidean space

Definition

Let $A$ , $B$ , $C$ , $D$ be four points of the Euclidean space $\R^3$ that do not lie in one plane. The tetrahedron $ABCD$ is the convex hull of $\set{A, B, C, D}$ . Its vertices are the four points, its edges are the six segments between them, and its faces are the four triangles on three of them.

Remarks

The four points do not lie in one plane exactly when the vectors $B - A$ , $C - A$ , $D - A$ are linearly independent. Then the volume of $ABCD$ is $V = \frac{1}{6} \abs{\det(B - A, C - A, D - A)}.$ A tetrahedron is a 3-dimensional simplex, and its four vertices always lie on one circumsphere. It is regular when all six edges have the same length; the regular tetrahedron is a Platonic solid.

Example: $(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1)$ . The determinant of the three standard basis vectors is $1$ , so the points are not coplanar and the volume is $1/6$ .

Non-example: $(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 0)$ all lie in the plane $z = 0$ . Their convex hull is a flat unit square, not a tetrahedron.

Reading path

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Convex hull concept
Euclidean space object
Plane concept
Tetrahedron definition

Tag	`R6PWCDFEVR`
URI	`m2://R6PWCDFEVR`
Kind	definition
Writing	W0 stub
MSC	51M20, 52B10
Defines	Tetrahedron
Uses	Convex hull, Plane, Euclidean space

Tetrahedron

Definition

Remarks

Reading path

Further reading

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