The proof of the volume of the truncated cone

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Discussion Overview

The discussion revolves around finding a proof for the volume of a truncated cone, focusing on the method rather than numerical examples. Participants explore various approaches and conditions under which the volume can be derived, including the case where the truncation is parallel to the base.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant requests a method to prove the volume of a truncated cone without numerical examples.
  • Another participant suggests that if the formula for the volume of a cone is known, deriving the volume of a truncated cone is trivial.
  • A participant notes that if the truncation is parallel to the base, the proof is straightforward, but it becomes more complicated otherwise.
  • One participant envisions removing the top of a full cone to conceptualize the proof and asks if there are alternative methods.
  • A detailed approach is presented involving setting up the truncated cone in a coordinate system and deriving the volume through integration of thin disks.
  • A participant inquires about using a deformation matrix to apply the regular formula when the truncation is not parallel to the base.
  • There is a clarification regarding the use of "dz" instead of "dx" in the integration process.

Areas of Agreement / Disagreement

Participants express differing views on the complexity of deriving the volume based on the orientation of the truncation. There is no consensus on a single method or approach, and the discussion remains unresolved regarding the best proof method.

Contextual Notes

Some assumptions about the conditions of the truncated cone are not fully explored, such as the implications of the truncation angle and its effect on the proof method.

Jasty
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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.
 
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Starting from where? If you can use the formula for volume of a cone, its trivial.
 
Jasty said:
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.

If the truncation is parallel to the base, Halls of Ivy is correct. Otherwise it is some what more complicated.
 
Yeah, I'll leave that to mathman!
 
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?
 
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 \pi [(R1-R2)z/H+ R2)^2 and volume \pi [(R1-R2)z/H+ R2)^2dx. To find the entire volume integrate that from z= 0 to z= H.
 
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ok thanks all.
 
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
 
HallsofIvy said:
Now imagine the entire cone, divided into thin disks: each has thickness "dz" and radius, x= (R1-R2)z/H+ R2 so area \pi [(R1-R2)z/H+ R2)^2 and volume \pi [(R1-R2)z/H+ R2)^2dx. To find the entire volume integrate that from z= 0 to z= H.

dz instead of dx?
 
  • #10
Yes, of course. Thanks.
 

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