SOLUTION: Donatello starts with a marble cube. He then slices a pyramid off each corner, so that in the resulting polyhedron, all the edges have the same side length. If the side lengt

Algebra ->  Percentage-and-ratio-word-problems -> SOLUTION: Donatello starts with a marble cube. He then slices a pyramid off each corner, so that in the resulting polyhedron, all the edges have the same side length. If the side lengt      Log On


   

Question 1205472: Donatello starts with a marble cube. He then slices a pyramid off each corner,
so that in the resulting polyhedron, all the edges have the same side length.
If the side length of the original cube is $6$, then find the volume of the resulting polyhedron.
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13209) About Me  (Show Source):
You can put this solution on YOUR website!


Each face of the resulting polyhedron is a regular octagon. View the regular octagon as a square with the corners cut off at 45 degree angles.

If s is the edge length of the resulting polyhedron, then the edge length of the cube (or side length of the square before the corners are cut off) is made up of one segment of length s and two segments of length s%2Fsqrt%282%29, for a total length of s%28sqrt%282%29%2B1%29.

Since the edge length of the original cube is 6, the edge length of the resulting polyhedron is



The volume of the resulting polyhedron is the volume of the original cube, minus the volume of the 8 pyramids cut off the corners.

The volume of the original cube is 6^3=216.

Each of the pyramids cut off the corners of the cube can be viewed as a pyramid with a base of an isosceles right triangle with side length s%2Fsqrt%282%29 and a height of s%2Fsqrt%282%29.

The area of the base of each of those pyramids is (one-half base times height)

%281%2F2%29%28s%2Fsqrt%282%29%29%5E2=s%5E2%2F4

The volume of each of the eight pyramids is (one-third base area times height)

%281%2F3%29%28s%5E2%2F4%29%28s%2Fsqrt%282%29%29=%28s%5E3%2Asqrt%282%29%29%2F24

The total volume of the eight pyramids is



So the volume of the resulting polyhedron is

216-%28720-504sqrt%282%29%29=504sqrt%282%29-504=504%28sqrt%282%29-1%29

Answer by ikleyn(52885) About Me  (Show Source):
You can put this solution on YOUR website!
.
Donatello starts with a marble cube. He then slices a pyramid off each corner,
so that in the resulting polyhedron, all the edges have the same side length.
If the side length of the original cube is 6, then find the volume of the resulting polyhedron.
~~~~~~~~~~~~~~~~~~~~~~~~


        This problem has two different solutions.
        One solution is just described by tutor  @greenestamps.
        In his solution,  the resulting polyhedron has  6  faces that are regular octagons
        with the side length of   6%2A%28sqrt%282%29-1%29   units.

        Another possible solution is when Donatello cuts off greater pieces of corners:
        greater pyramids with three orthogonal edges of  3 = 6/2  units each.

        Then the resulting polyhedron has  6  faces that are  SQUARES
        with the side length of   3%2Asqrt%282%29   units each. 


Then the volume of each cut off pyramids is  

     V = %281%2F3%29%2Abase%2Aheight = %281%2F3%29%2A%28%281%2F2%29%2A3%5E2%29%2A3 = 9%2F2 cubic units,


and the volume of the remaining polyhedron is  6%5E3+-+8%2A%289%2F2%29 = 216 - 4*9 = 216 - 36 = 180 cubic units.


ANSWER.  The second solution provides the polyhedron of the volume 180 cubic units.