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.
Thanks!
+23245 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.
