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  "body": "\n## Definition\n\nLet $A$, $B$, $C$, $D$ be four points of the [[Euclidean space|0XPR1E85W0]] $\\R^3$ that do not lie in one [[plane|0V3XAMF3C1]].\nThe *tetrahedron* $ABCD$ is the [[convex hull|0FQSDC2EQ2]] of $\\set{A, B, C, D}$.\nIts vertices are the four points, its edges are the six segments between them, and its faces are the four triangles on three of them.\n\n## Remarks\n\nThe four points do not lie in one plane exactly when the vectors $B - A$, $C - A$, $D - A$ are linearly independent.\nThen the volume of $ABCD$ is\n$$V = \\frac{1}{6} \\abs{\\det(B - A, C - A, D - A)}.$$\nA tetrahedron is a 3-dimensional [[simplex|YX1DJK2QAG]], and its four vertices always lie on one [[circumsphere|MRC4WMVTAH]].\nIt is regular when all six edges have the same length; the regular tetrahedron is a [[Platonic solid|017E5PRS1Q]].\n\nExample: $(0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1)$.\nThe determinant of the three standard basis vectors is $1$, so the points are not coplanar and the volume is $1/6$.\n\nNon-example: $(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 0)$ all lie in the plane $z = 0$.\nTheir convex hull is a flat unit square, not a tetrahedron.\n",
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