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Cone Frustum


Date: 08/31/2001 at 13:47:43
From: Mitch Rosewall
Subject: Frustum Cone

I am trying to calculate the height of a right frustum cone knowing 
only r, V, and the angle of the side. These are the only constants I 
have. Can you help me?


Date: 08/31/2001 at 15:43:55
From: Doctor Rob
Subject: Re: Frustum Cone

Thanks for writing to Ask Dr. Math, Mitch.

I drew this picture of a vertical cross-section of the cone and its
frustum:

                V
                o
               /|\
              / | \
             /  |  \
            /  x|   \
           /    |    \
        D o-----o-----o F
         /   r  |E     \
        /       |       \
       /       h|        \
      /         |         \
     /A    R    |          \
    o-----------o-----------o
   A            C            B

You know R, the measure of <A, and the volume V.

Now trigonometry and the formula for the volume come to our aid:

   tan(A) = (x+h)/R = x/r,
   x = R*tan(A) - h,
   r = R - h*cot(A).

Then

   V = (R^2+R*r+r^2)*Pi*h/3,
   V = (R^2+R*[R-h*cot(A)]+[R-h*cot(A)]^2)*Pi*h/3.

Now this is a cubic equation in h in which all the coefficients are
are known from your given data.  It can be put into the form

   h^3 + h^2*[-3R*tan(A)] + h*[3*R^2*tan^2(A)] +
                                             [-3*V*tan^2(A)/Pi] = 0.

To solve cubic equations, see the following web page from our
Frequently Asked Questions (FAQ):

   Cubic and Quartic Equations
   http://mathforum.org/dr.math/faq/faq.cubic.equations.html   

This equation is particularly simple to solve, and the only real
root has the value

   h = R*tan(A) - (R^3*tan^3[A]-3*V*tan^2[A]/Pi)^(1/3).

That formula is the answer to your question.

Example:  

If R = 10 cm, V = 500 cm^3, and m(<A) = 60 degrees, then 
tan(A) = sqrt(3), and so

   h = 10*sqrt(3) - (3000*sqrt[3]-4500/Pi)^(1/3),
     = 1.76537 cm.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   


Date: 09/03/2001 at 14:20:05
From: Mitch Rosewall
Subject: Calculating "R" for Frustum

You have helped me find r, but What I need is to find R and H when 
r, the volume of the frustum, the and angle of the side are known.  



         /|-----r------\
        / |             \
       /  H              \
      /   |               \
     /    |                \
    /----------R------------\ 

Please help me.

Mitch


Date: 09/04/2001 at 10:57:05
From: Doctor Rob
Subject: Re: Calculating "R" for Frustum

Thanks for writing back, Mitch.

The question I answered was how to find H given R, the volume, and the
angle.

If you know r, V and the angle A, the previous answer contained the
following formulas:

   tan(A) = (x+H)/R = x/r,
   x = R*tan(A) - H,
   r = R - H*cot(A),
   H = (R-r)*tan(A).

Then

   V = (R^2+R*r+r^2)*Pi*H/3,
     = (R^2+R*r+r^2)*Pi*(R-r)*tan(A)/3,
     = (R^3-r^3)*Pi*tan(A)/3.

Now this is a cubic equation in R in which all the coefficients are
are known from your given data.  It can be put into the form

   R^3 = r^3 + 3*V*cot(A)/Pi.

The only real root has the value

   R = (r^3+3*V*cot[A]/Pi)^(1/3).

That formula is the answer to your first question.

Once you know R, you can find

   H = (R-r)*tan(A),
     = -r*tan(A) + (r^3*tan^3[A]+3*V*tan^2[A]/Pi)^(1/3).

That formula is the answer to your second question. 

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

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