Results 1 to 2 of 2

Math Help - ratio of volumes

  1. #1
    Junior Member
    Joined
    Sep 2008
    Posts
    62

    ratio of volumes

    Show that the ratio of the volumes of two similar pyramids
    is equal to the ratio of the cubes of corresponding edges- not using modern
    calculus,but by comparing magnitudes
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Apr 2005
    Posts
    1,631
    Pyramid.
    V = (1/3)(area of base)(perpendicular height of the apex from the base)

    area of base is in square units.
    If all its sides are "a", then area = k*a^2, where k is a constant.
    If its sides are in "a" and "b", the area = j*a*b, where j is a constant. But it can be solved that b = m*a, where m is a constant, so area = n*a^2, where n is a constant.
    If the base has many edges, it can be shown that the area will be p*a^2, where p is a constant and "a" is one of the edges of the base

    The h, or perpendicular height of the apex.
    It can also be solved that h is proportional to the "a" in the base. So, h = r*a, where r is a constant.

    Therefore, the volume of the pyramid, in terms of "a" only is
    V = (1/3)(p*a^2)(r*a)
    V = (pr/3)(a^3) ------remember that p and r are constants, so the unit of the volume is still cubic.

    Now we get two similar pyramids, where "a" is an edge of the base of one, and "b" is an edge of the base of the other, and that "a" and "b" are corresponding edges of the two similar pyramids.

    V1 = (pr/3)(a^3)
    V2 = (pr/3)(b^3)

    V1 / V2
    = [(pr/3)(a^3)] / [(pr/3)(b^3)]
    = a^3 / b^3

    Shown.

    Even on any corresponding slant edges of the similar pyramids it can be shown just the same.
    Last edited by ticbol; September 12th 2008 at 05:31 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Ratio of Volumes of a cone
    Posted in the Geometry Forum
    Replies: 5
    Last Post: May 15th 2011, 09:16 AM
  2. Ratio of Volumes
    Posted in the Geometry Forum
    Replies: 1
    Last Post: February 27th 2010, 02:31 AM
  3. Replies: 6
    Last Post: February 11th 2010, 08:08 PM
  4. Ratio for curve and point-ratio form
    Posted in the Algebra Forum
    Replies: 4
    Last Post: January 11th 2010, 01:07 AM
  5. Finding n given the ratio of nCr to another ratio
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: October 5th 2009, 08:03 PM

Search Tags


/mathhelpforum @mathhelpforum