Let \(A_1 A_2 A_3 A_4\) be a regular tetrahedron. Let \(P_1\) be the center of face \(A_2 A_3 A_4,\) and define vertices \(P_2, P_3,\) and \(P_4\) the same way. Find the ratio of the volume of tetrahedron \(A_1 A_2 A_3 A_4\) to the volume of tetrahedron \(P_1 P_2 P_3 P_4.\)
Note: A tetrahedron is called regular if all the edges lengths are equal, so all the faces are equilateral triangles.
Since each of the points Pi is the centroid of the triangle that includes it, it is found one-third from the distance from the base to the vertext of this triangle.
Therefore, the height of the inner tetrahedron is 1/3rd the height of the outer tetrahedron.
Since these shapes are similar, the volume of the inner tetrahedron is (1/3)3, or one-twenty-seventh the volume of the outer tetrahedron.
This makes the volume of the outer tetrahedron 27 times the volume of the inner tetrahedron.