University of Georgia
EMAT 6690
Claudette Tucker
Essay Six
InterMath: Octahedron in a Box
Investigation: Recall the volumes of a cube and an octahedron.
Volume of a cube = 
Volume of an octahedron = 
For this investigation, we must determine the probability of choosing a point inside of an octahedron that is inscribe in a cube. The cube and the octahedron are dual polyhedron, meaning that we take either object and transform it into the other. Here the faces of the cube transform into vertices and edges change to faces. Notice that each vertex on the octahedron is the center of the faces of the cube. So, we can cut the cube along two vertices of the octahedron to help to determine the probability as such.
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Triangle GHC forms a right triangle. Therefore, we can use the Pythagorean Theorem or the distance formula to find the length of the hypotenuse. This value will be used to calculate of the volume of cube.
Distance Formula
So, the volume of the octahedron is equal to
and the volume
of the cube is one.
So, to calculate the probability of randomly selecting a
point in the cube and octahedron, we first must compare volume of each space,
octahedron to cube. So, we have,
the total volume of the octahedron divided by the total volume of the cube, 
. This means
that the octahedron occupies seventeen percent of the cubeÕs volume. It also reveals that seventeen percent
of the point selected will land inside the cube and octahedron or out of every
six random points only one will land in the octahedron.