SOLUTION: All the corners are cut off a cube. These cuts are just large enough so that the new triangular faces tough at their vertices. (cuboctahedron). All the edges are then the same leng

Algebra ->  Volume -> SOLUTION: All the corners are cut off a cube. These cuts are just large enough so that the new triangular faces tough at their vertices. (cuboctahedron). All the edges are then the same leng      Log On


   

Question 1118068: All the corners are cut off a cube. These cuts are just large enough so that the new triangular faces tough at their vertices. (cuboctahedron). All the edges are then the same length. If all the edges have length 1cm, what is the exact volume of the solid?
Found 2 solutions by ikleyn, KMST:
Answer by ikleyn(52800) About Me  (Show Source):
You can put this solution on YOUR website!
.
The cube has 8 corners, so there are 8 cuts.


Each the cut is a pyramid with three right angled faces.


The orthogonal edges  of the pyramids are 1%2Fsqrt%282%29 = sqrt%282%29%2F2 centimeters long each.


So the volume of each pyramid is  V = %281%2F3%29%2A%281%2F2%29%2A%28sqrt%282%29%2F2%29%5E3 = %281%2F3%29%2A%281%2F2%29%2A%28%282%2Asqrt%282%29%29%2F8%29 cubic centimeters.


The total volume of 8 pyramids is  %281%2F3%29%2A%28sqrt%282%29%29 cubic centimeters.


The edge of the cube is 2%2A%28sqrt%282%29%2F2%29 = sqrt%282%29 centimeters.


So the volume of the cube is  %28sqrt%282%29%29%5E3 = 2%2Asqrt%282%29 cubic centimeters.


Then the volume of the solid after cutting the corners of the cube is the difference


2%2Asqrt%282%29 - %281%2F3%29%2Asqrt%282%29 = %285%2F3%29%2Asqrt%282%29 cubic centimeters.


It is your answer.

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On  cuboctahedron,  see this  Wikipedia  article

https://en.wikipedia.org/wiki/Cuboctahedron

You will find the Figure there.



Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Wikipedia (and other sources) would give you a formula for the volume of a cuboctahedron as a function of the edge length,
but formulas are for computers, who cannot think for themselves.
Humans are supposed to be able to think using their own brains.

The volume of the cuboctahedron is the volume of the original cube minus the volume of the parts cut off.
To cut the corners that way,
you will cut along lines connecting midpoints of adjacent edges.
Each of the six faces of the cube will be cut,
leaving a square face for the cuboctahedron,
and losing four corner triangles, like this:
.
The black lines are the 1-cm edges of the cuboctahedron,
which has 6%2A4=24 such edges separating
6 square faces that are portions of the cube faces
from 8 triangular "freshly cut" faces.
The Pythagorean theorem tells us that
the length of the green sides of those "lost" isosceles right triangular pieces of each cube face is
green%28x%29cm such that green%28x%29%5E2%2Bgreen%28x%29%5E2=1 , so green%28x%29=green%28sqrt%28%221+%2F+2%22%29%29=green%28sqrt%282%29%2F2%29 ,
and the length (in cm) of the edge of the cube is green%282x%29=green%28sqrt%282%29%29 .

The volume of the cube, before it was butchered) was
%28green%28sqrt%282%29%29%29%5E3=green%282sqrt%282%29%29 cubic cm .

As for the pieces cut off,
the 4 that were resting on the table are still there,
sitting on a right triangular face.
If you rearrange them, joining all right angles,
you get a nice square-base Egyptian-style pyramid,
with side length 1cm , and height green%28sqrt%282%29%2F2%29cm .
You can do the same with the 4 corners cut off from the top face of the cube.
Those two pyramids, if you join them by the square bases form an octahedron.
The volume of that octahedron (2 pyramids), in cubic cm, is
.

So, you have an octahedrom made from the cut-off pieces,
and a cuboctahedron whose volume, in in cubic cm, is
.