Results 1 to 5 of 5

Math Help - Volume of Pyramid..

  1. #1
    Newbie
    Joined
    Jan 2010
    Posts
    3

    Volume of Pyramid..

    Hi there, I have a homework question that asks why the volume of a pyramid is 1/3BH... I'm not even sure why it is.. I've just always been taught that formula and how to use it. Any ideas?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Jun 2009
    Posts
    806
    Thanks
    4
    Let R be each side of the base and H is the height. Take a slice of the pyramid of thickness dh and length r at a height h from the vertex.
    The volume of this slice is
    dV = r^2*dh.By simple geometry you can see that R/H = r/h.
    Hence
    r = (R/H)*h.
    dV = (R/H)^2*h^2*dh.
    To find the total volume find the integration between h = 0 to h = H.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jan 2010
    Posts
    3
    Ok i think i'm understanding a little.. but why is it 1/3 and not 1/2?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Jun 2009
    Posts
    806
    Thanks
    4
    Quote Originally Posted by Schmu02 View Post
    Ok i think i'm understanding a little.. but why is it 1/3 and not 1/2?
    V = Intg(R^2/H^2)*h^2*dh = (R^2/H^2)*h^3/3.
    When you substitute h = H,
    V = 1/3*R^2*H. Bur R^2 = B = area of the base.
    So V = 1/3*BH.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Hello Schmu02
    Quote Originally Posted by Schmu02 View Post
    Hi there, I have a homework question that asks why the volume of a pyramid is 1/3BH... I'm not even sure why it is.. I've just always been taught that formula and how to use it. Any ideas?
    Are you supposed to use a calculus method here? If so, sa-ri-ga-ma's solution is fine. If not, take a look at the diagram I've attached.

    This is a cube ABCDEFGH, with lines joining all the vertices to a single vertex, A.


    If you look very hard, you'll see that the cube is then made up of three identical pyramids, each with its vertex at A. The three pyramids have bases:
    BFGC, EFGH and CGHD
    The volume of each pyramid is therefore one-third the volume of the cube.

    Now whatever pyramid you start with, you can always find a pyramid with the same base area and the same height that looks just like one of these three*. Its volume is therefore one-third base area x height.

    Grandad

    * P.S. I don't think that's strictly true, because we've started with pyramids whose base area is equal to the square of their height. So, you may need to stretch (or compress) the pyramid a bit along its altitude. But the principle remains the same.
    Attached Thumbnails Attached Thumbnails Volume of Pyramid..-untitled.jpg  
    Last edited by Grandad; March 23rd 2010 at 08:29 AM. Reason: Add PS
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Pentagonal Pyramid Volume help
    Posted in the Geometry Forum
    Replies: 3
    Last Post: November 14th 2010, 08:20 AM
  2. Volume of a Pyramid
    Posted in the Calculus Forum
    Replies: 2
    Last Post: January 27th 2010, 08:08 PM
  3. volume of a pyramid using integration
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 28th 2009, 11:37 AM
  4. Volume of a pyramid
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 12th 2009, 08:26 PM
  5. Volume of a pyramid
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 1st 2009, 07:57 PM

Search Tags


/mathhelpforum @mathhelpforum