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#1 2013-05-27 23:32:54

{7/3}
Member
Registered: 2013-02-11
Posts: 210

Pyramid volume

A isoceles triangle base pyramid's apex is on top of the triangle's top vertex,what is it's volume?is there a formula?how was it derived?


There are 10 kinds of people in the world,people who understand binary and people who don't.

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#2 2013-05-28 00:02:28

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,180

Re: Pyramid volume

Hi {7/3};

I am not getting the question. Do you have a sketch?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#3 2013-05-28 02:16:53

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 16,579
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Re: Pyramid volume

He actually wants to ask about a general formula for a pyramid

Which is Area of the base * Height * 1/3

Last edited by Agnishom (2013-05-28 02:19:15)


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

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#4 2013-05-28 03:20:56

bob bundy
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Registered: 2010-06-20
Posts: 6,374

Re: Pyramid volume

hi {7/3}

That works for any 'pyramid', whatever the shape of the base.

Many mathematicians insist that to be a pyramid, the base should be a polygon, but for the following proof, let's make the base any shape at all. (see picture)  Then to be a pyramid like that, all I require is that any length, d,  at distance x from the vertex and the corresponding distance in the base, D, are related by this formula:

where H is the height of the pyramid.

Because the dark blue base region is mathematically similar to the light blue region  the areas, small:large, will be in this proportion:

Now for a little integral calculus:

Imagine the solid divided into thin slices parallel to the base with thickness delta x.

So let the slices become infinitesimally thin with thickness 'dx', and sum all slices from the top to the base

Furthermore, the pyramid need not be 'right', ie. the axis at right angles to the base.  If a right pyramid is sheared parallel to the base, so that it leans over, the slices still have the same area, so the volume formula continues to work, provided H is the perpendicular height.  (picture in next post)

Bob

View Image: volume of pyramid.gif

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#5 2013-05-28 03:37:32

bob bundy
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Registered: 2010-06-20
Posts: 6,374

Re: Pyramid volume

Non-right pyramid

B

View Image: volume of pyramid2.gif

You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#6 2013-05-28 04:10:35

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 16,579
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Re: Pyramid volume

Thanks
This is what I was eager for.

I still want to know, how did you get:


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

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#7 2013-05-28 05:04:10

{7/3}
Member
Registered: 2013-02-11
Posts: 210

Re: Pyramid volume

Thanks,is there any proof without calculus,i'm just curious.


There are 10 kinds of people in the world,people who understand binary and people who don't.

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#8 2013-05-28 05:06:30

bob bundy
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Registered: 2010-06-20
Posts: 6,374

Re: Pyramid volume

hi Agnishom,

The two blue shapes are similar.  So there is a scale factor that transforms lengths in the large shape into the corresponding lengths in the small shape.

The area scale factor in such a situation is the square of the length scale factor.

http://www.mathsisfun.com/geometry/tria … orems.html

section 3.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#9 2013-05-28 05:14:16

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,374

Re: Pyramid volume

hi {7/3}

I've had a quick search on-line and found two that attempt this.  But both use the underlying ideas of calculus whilst avoiding the formal integral approach.  The nrich pages are full of interesting maths so it's well worth a look.

http://mathforum.org/library/drmath/view/55041.html

http://nrich.maths.org/1408

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#10 2013-05-28 05:45:58

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 16,579
Website

Re: Pyramid volume

Thanks


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

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