000RM.dvi

42.3 Volume of a tetrahedron 1109

42.3 Volume of a tetrahedron

LetABCDbe a tetrahedron with

BC=a, CA=b, AB=C,

AD=a′,BD=b′,CD=c′.

The volumeVof the tetrahedron is given by

V^2 =

1

288

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

01 1 1 1

10 a′^2 b′^2 c′^2

1 a′^20 c^2 b^2

1 b′^2 c^20 a^2

1 c′^2 b^2 a^20

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ =

1

144

((

∑

edges

a^2

)

(∑

a^2 a′^2

)

− 2

∑

a^2 a′^2 (a^2 +a′^2 )−

∑

faces

a^2 b^2 c^2

)

.