The cube's edges have been cut off. How to calculate the volume of what remains?

[Ana.stasia]

The question is:
The cube with a side a had all of its edges cut off so that each side of the cube has become an octagon with all sides equal. Calculate the volume of what remains of the cube.

The solution must be calculate through a with no other symbol involved.

Now it's clear to me that to get the volume of what remain I need to calculate the volume of the eight edges that have been cut off (those edges being pyramids) and subtract that from the volume of the cube.

This is the approach I tried to take.

V_e is the volume of what remains.






Now here is where I encountered a problem. I proposed that what I marked as c and what I marked as x are not the same because If they were the same then a=3x and x=a/3 when the edge of the pyramid is supposed to be a/2 * (2- square root of 2) unless the solution in the book is not correct. Also the next problem I encountered is that I proposed that a= x+ 2c, are the two Cs I marked possibly not the same? or do they have to be the same because octagon has all equal sides.



Here is how I tried to calculate the height. Now I am fully aware that my x does not the match the x from the solution, but I decided to try to calculate It until the end because a/2*(2- square root of 2) = a - a*square root of 2/2 which is very similar to what I got so I didn't rule out the possibility of a typo. The volume shown in the solution also includes *(square root of 2 - 1), so I decided to still try with the x I got.



This is the calculation of a volume which could possibly be shorten, however given I decided to leave it at this unless I am sure it's correct.

What mistake did I make in my calculations?

Thank you in advance

 

[Dr.Peterson]

It appears that what you are calling "edges" are "vertices", or "corners". A cube has 12 edges, and 8 vertices.

Also, what you call x appear to be the edges of the equilateral triangle that forms one of 8 faces of the resulting solid (and therefore also any of the 8 equal edges of the octagonal face), while c is the portion of the edge of the cube that is cut off. Am I right?

Then you are right that [MATH]x = a(\sqrt{2}-1)[/MATH], and [MATH]c = \frac{x\sqrt{2}}{2}[/MATH]. Certainly it would make no sense that x = c; c is not a side of the octagon: x is.

Are you saying that the book says [MATH]x = \frac{a}{2} \left(2- \sqrt{2}\right)[/MATH] ? That's what 2c is.

But to find the volume of one of the 8 pieces cut off, I would not take the equilateral triangle as the base, as the height from that base is relatively difficult to determine. Instead, I would take as base one of the right isosceles triangles with sides c, and then the height would just be c. Do you see that?

I didn't read through the last two images, which are far too complicated. At the least, you should factor [MATH]a^3[/MATH] out of all volume expressions; and you should not have [MATH]a^2[/MATH] in the expression for H!

But I think if you take my approach to the volume, you should find it much easier.

 

[Ana.stasia]

It appears that what you are calling "edges" are "vertices", or "corners". A cube has 12 edges, and 8 vertices.

Also, what you call x appear to be the edges of the equilateral triangle that forms one of 8 faces of the resulting solid (and therefore also any of the 8 equal edges of the octagonal face), while c is the portion of the edge of the cube that is cut off. Am I right?

Then you are right that [MATH]x = a(\sqrt{2}-1)[/MATH], and [MATH]c = \frac{x\sqrt{2}}{2}[/MATH]. Certainly it would make no sense that x = c; c is not a side of the octagon: x is.

Are you saying that the book says [MATH]x = \frac{a}{2} \left(2- \sqrt{2}\right)[/MATH] ? That's what 2c is.

But to find the volume of one of the 8 pieces cut off, I would not take the equilateral triangle as the base, as the height from that base is relatively difficult to determine. Instead, I would take as base one of the right isosceles triangles with sides c, and then the height would just be c. Do you see that?

I didn't read through the last two images, which are far too complicated. At the least, you should factor [MATH]a^3[/MATH] out of all volume expressions; and you should not have [MATH]a^2[/MATH] in the expression for H!

But I think if you take my approach to the volume, you should find it much easier.

I applied the method you gave me and got this solution.



It doesn't match the solution from the book only by a number. That is I divide everything by 3, and the solution has a 6 instead of 3.



However I do not see a mistake in my work regarding that. Did I miss something or is this a typo?

 

[Dr.Peterson]

You agree with their unsimplified version; so the difference is in the simplification. And it appears that they are wrong:

1-8/6*((2-sqrt(2))/2)^3 - Wolfram|Alpha

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