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The proof of the volume of the truncated cone

  1. Apr 30, 2008 #1
    Please I need a respectable proof how to get the volume of the truncated cone. I need it really quick. So please could you help me. No numbers just "the method" how to get that formula. Thanks.
     
  2. jcsd
  3. Apr 30, 2008 #2

    HallsofIvy

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    Starting from where? If you can use the formula for volume of a cone, its trivial.
     
  4. Apr 30, 2008 #3

    mathman

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    If the truncation is parallel to the base, Halls of Ivy is correct. Otherwise it is some what more complicated.
     
  5. Apr 30, 2008 #4

    HallsofIvy

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    Yeah, I'll leave that to mathman!
     
  6. May 1, 2008 #5
    I mean the most basic conditions. I can imagine the whole cone and than remove the top. Is there any other way how to prove it?
     
  7. May 1, 2008 #6

    HallsofIvy

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    Let C be a truncated, right circular cone with height H, upper radius R1 and lower radius R2. Set it up on a coordinate system with the center of the base at (0,0,0), and center of the top at (0,0,H). Looking at it from the side, so that you see the xz-plane, you see a "trapezoid" with one side starting at (R2,0,0) and ending at (R1,0,H). Since any nonvertical line in the xz-plane can be written in the form z= Ax+ B. you must have 0= R2A+ B and H= R1A+ B. Subtracting the first from the second, H= (R1-R2)A so A= H/(R1-R2), B= -R2A so B= -R2H/(R1-R2). The equation of the line is z= H(x-R2)/(R1-R2) or you can write it x= (R1-R2)z/H+ R2.

    Now imagine the entire cone, divided into thin disks: each has thickness "dz" and radius, x= (R1-R2)z/H+ R2 so area [itex]\pi [(R1-R2)z/H+ R2)^2[/itex] and volume [itex]\pi [(R1-R2)z/H+ R2)^2dx[/itex]. To find the entire volume integrate that from z= 0 to z= H.
     
    Last edited: Feb 18, 2010
  8. May 4, 2008 #7
    ok thanks all.
     
  9. May 4, 2008 #8
    Can we do this by using a deformation matrix and then using the regular formula if it is not parallel to the base? Just out of curiosity, couldn't see it right away
     
  10. Feb 18, 2010 #9
    dz instead of dx?
     
  11. Feb 18, 2010 #10

    HallsofIvy

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    Yes, of course. Thanks.
     
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