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Not true three edges !!!
Not true three edges !!!
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since when did pythagoras take into account arcs ?
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It works out to calculating the volume of a thee dimensional triangle (pyramid) where on one plane only it incoporates a curved hypotenuse and adjacent side (or face). Then multiply the answer by twelve then add this to the volume of the cube minus the volume of the three sided pyramid x 2.
Easy !
It's essentially a geometry question. Break it down to manageable shapes.
Easy !
It's essentially a geometry question. Break it down to manageable shapes.
C
calorific
Pardon 
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Its the way to work out the answer !
You asked the question !
Just re read it !
Ok you drill the hole though from one corner to the opposite, calculate the lowest point at which the bored hole meets the face of one of the sides, easy enough.
Then calculate the volume of the pyramid at the lowest point of this arc.
take this volume from the total volume of the cube x2.
So now all you need to do is add the twelve segments that you've taken off for the purposes of the calculation (two on each face as they are symmetrical. )
The shape you have to add is a pyramid with a concave face so all that's needed is to calculate this one shape multiply by twelve and thats the answer. On a drawing board I could do this with a planometer by sectioning the shape and calculate it that way.
Although Ive sketched the shape I don't have the formula to calculate it,
If I had a planometer and drawing board I could give you the answer !
You asked the question !
Just re read it !
Ok you drill the hole though from one corner to the opposite, calculate the lowest point at which the bored hole meets the face of one of the sides, easy enough.
Then calculate the volume of the pyramid at the lowest point of this arc.
take this volume from the total volume of the cube x2.
So now all you need to do is add the twelve segments that you've taken off for the purposes of the calculation (two on each face as they are symmetrical. )
The shape you have to add is a pyramid with a concave face so all that's needed is to calculate this one shape multiply by twelve and thats the answer. On a drawing board I could do this with a planometer by sectioning the shape and calculate it that way.
Although Ive sketched the shape I don't have the formula to calculate it,
If I had a planometer and drawing board I could give you the answer !
C
calorific
Marvellous. Any chance you elaborating a little bit on the nature of the curve of the base of this triangle, like maybe giving some kind of equation for it? Or is such a thing so obvious to you that it's beneath contempt to ask?
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No need for sarcasm you posted a serious question and I was interested to see if I could work it out.
Had I been in one of my previous jobs I would have had the equipment to give a pretty accurate answer.
The three dimensional pyramid with a curved face is proving difficult, image doesn't come up on google, and I cant find a formula to calculate the volume.
Had I been in one of my previous jobs I would have had the equipment to give a pretty accurate answer.
The three dimensional pyramid with a curved face is proving difficult, image doesn't come up on google, and I cant find a formula to calculate the volume.
C
calorific
Sorry, wasn't meaning to be sarcastic an a nasty way
It's just the curve at the bottom of the pyramid is key to the whole thing, and is still being quite elusive
It's just the curve at the bottom of the pyramid is key to the whole thing, and is still being quite elusive
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There is no curve at the bottom of the pyramid, you have a pyramid where one face only is caused by the arc of the drill and the hypotenuse is curved as it reaches the centre line of the face (where for the calculation you stop). Thats why you then multiply by six for the total volume at each end, then double it.
I know the shape we've got but cant find a formula, might have to did out some of my old books.
I know the shape we've got but cant find a formula, might have to did out some of my old books.
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Imagine you cut a slice of cake, you then lay it flat on one face, you then cut from the point of what would have been the centre at the bottom to the what would have been top outside edge.
you then cut an arc giving you a concave surface to one of the faces.
As Ive already said drawing board and planometer ........easy !
you then cut an arc giving you a concave surface to one of the faces.
As Ive already said drawing board and planometer ........easy !
Well done Sherlock!
If you cut through a flat plane with a circular hole this makes the hole eliptical that means each surface of the cube would have a portion of an elips
This is so easy, go out and chop down a tree, next cut it down to size, leaving a bit extra for sandpapering, when you have your little cube drill a 1mm hole through the longest opposit corners then just drop it into a eureka glass, simples
Well...
What I said was that pythagoras was 'as hard as it gets'.
And in any event, Pythagoras is more than just a simple equation, it can be used to derive all sorts of things, including stuff that includes arcs. The beauty of such a simple equation is it's power.
Pythagoras can be used to calculate the radius of the inscribed circle of a triangle, and also the radius of the circumscibed circle of a triangle.
http://en.wikipedia.org/wiki/Equilateral_triangle#Principal_properties
Both these calculations need to be made to solve this problem. The two circles are key to working out the height of that strange shape you go on about. The location of the excircle also allows us to calculate the amount of wood lost as the drill penetrates the square at the corner.
(I don't think it's an equilateral triangle but the principle applies, still pythagoras)
Once you have worked out the height (h) of the shape, you need to calculate the area of the base (that's the incircle plane worked out previously) of your funny shape. The area of the base is simply the area of the triangle at that plane minus the csa of the drill bit. As there are three funny shapes you divide that answer by three, giving a.
You then apply the standard formula for the volume of a cone. base x height / 3 or in our example ah/3
That's the hard part done.
Then a bit more simple geometry and bob's your uncle.
All pythagoras apart from calculating the area of a circle (drill bit), you just need to break the square down into a load of right angle triangles to work it out.
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