5. Find the volume of an octahedron whose edges are all 1 m.
An octahedron consists of two square pyramids pasted together at their square bases. If we could find the volume of the top pyramid, we would need only double it to find the total volume. The volume of a pyramid is
The base is a square whose edges are all 1 m, so the area of the base is 1m2. We need only find the height. The height is part of the following right triangle
O is the center point of the octahedron, T is the top and B is one of the vertices of the base. BT is one of the edges of the octahedron so it is 1 m. OB is half of a diagonal of a square whose edges are all 1 m, so it is half of the square root of 2 meters. We cah now use the Pythagorean Theorem to find h.
This verifies that the line from the top of the figure to the bottom verex is a diagonal of a square. The volume of the octahedron is