Art of Problem Solving AoPS Online Beast Academy AoPS Academy 2006 SMT/Geometry Problems/Problem 1 Problem Given a cube, determine the ratio of the volume of the octahedron formed by connecting the centers of each face of the cube to the volume of the cube. Solution Let the side length of the square be . Consider Thus, . It's an isosceles right triangle with hypotenuse , and the side length of the octahedron is and legs of length . . Now consider the top half of the octahedron. It's a pyramid with a square base of length and height , and the volume of the entire octahedron is twice this, or Finally, the ratio of the volume of the octahedron to the volume of the cube is . Therefore, its volume is . . See Also 2006 SMT/Geometry Problems Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2006_SMT/Geometry_Problems/Problem_1&oldid=47209" Copyright © 2023 Art of Problem Solving
