Question

A tetrahedron is a triangular pyramid in which each face is an equilateral triangle. Calculate the volume of a tetrahedron that has all its edges 6 cm in length. Hint: You will need to know some geometry and trigonometry.

Answer 1

the height of the tetrahedron will hit the centroid of the base.
and will be 2/3 of the way from the vertex.
Length of median of base:
m^2 + 3^2 = 6^2
m= √27 =3√3
so distance from vertex to centroid = (2/3)(3√3) = 2√3
let the height of the pyramid be h
h^2 + (2√3)^2 = 6^2
h = √24 = 2√6

area of base = (1/2)(6)(6)sin60°
= 18√3/2 = 9√3

volume of tetrahedron
= (1/3) base x height
= (1/3)(9√3)(2√6) = 18√2