(Volume of a tetrahedron) A tetrahedron is a pyramid with a
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(Volume of a tetrahedron) A tetrahedron is a pyramid with a triangular base and three other triangular faces. It has fou
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(Volume of a tetrahedron) A tetrahedron is a pyramid with a triangular base and three other triangular faces. It has four vertices and six edges. Like any pyramid or cone, its volume is equal to 1 3Ah, where A is the area of the base and h is the height measured perpendicular to the base. If u, v, and w are vectors coinciding with the three edges of a tetrahedron that meet at one vertex, show that the tetrahedron has volume given by Volume D 1 6 ju .v w/j D 1 6 j ? ? ? ? ? ? u1 u2 u3 v1 v2 v3 w1 w2 w3 ? ? ? ? ? ? j: Thus, the volume of a tetrahedron spanned by three vectors is one-sixth of the volume of the parallelepiped spanned by the same vectors.
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