Find the volume 𝑉 of a regular tetrahedron whose face is an equilateral triangle of side 𝑠.

Question

the tetrahedron has height √2/3s

Find the volume of the tetrahedron for 𝑠=6.

Answer 1

The base of the tetrahedron is an equilateral triangle with area

B = (sqrt(3) / 4) s^2

As you have noted, the tetraheedron has height h = sqrt(2 / 3) s.

So the tetrahedron has volume

V = (1 / 3) B h = (sqrt(2) / 12) s^3.

In particular, when s = 6, we have V = 18 sqrt(2) ≈ 25.46.