Volume of Tacoma Dome

Date: 01/16/2002 at 18:54:19
From: Kacey Beyers
Subject: Volume of Tacoma Dome

I am trying to figure the volume of the Tacoma Dome in Tacoma, 
Washington. I know the Diameter is 530'. I know the height to the 
tallest point of the dome is 152'. The side wall all around the dome 
is 36' high. 

I got as far as finding the volume of the base of the dome, which is 
pi r2 h.  

= pi (265)(265) h
= pi 70225 x 36
= pi x 2528100

The problem I'm having is finding the volume of the lid of the dome.  
Since it's not a full sphere, I don't really know where to begin.  
Please give me some advice.

Thanks, 
Kacey

Date: 01/17/2002 at 02:42:33
From: Doctor Jeremiah
Subject: Re: Volume of Tacoma Dome 

Hi Kacey,

In this answer I am going to use * for multiply to avoid confusion 
with the letter x, and I will use ^ to mean "an exponent of" so x^2 
would be x squared.

The shape you're talking about is called a 'spherical cap'.

If I understand what you are asking, it is:

  What is the volume of a spherical cap of height 152 - 36 = 116, 
  whose base has a radius of 530 (where the radius of the sphere 
  is much bigger)?

To determine this we need to find R (the sphere's radius). Consider 
this semi-circle:

                   530 feet
             |-----------------|
                    +++++             -+-
               +++         +++         | 152 - 36 = 116 feet
           +++---265--+        +++    -+-
        +     \       |       /     +
      +        \      |      /        +
     +          \     |     /          +
    +            R    H    R            +
   +              \   |   /              +
   +               \  |  /               +
  +                 \ | /                 +
  +                  \|/                  +
  +                   +                   +


Basically we have a triangle, and Pythagoras' theorem says:

  R^2 = H^2 + 265^2

and since H = R-116 we have:  

        R^2 = (R - 116)^2 + 265^2

        R^2 = (R^2 - 232R + 13456) + 70225

  R^2 - R^2 = -232R + 83681

          0 = -232R + 83681

       232R = 83681

          R = 83681/232 feet


          H = 83681/232 - 116

          H = 83681/232 - 26912/232

          H = 56769/232 feet

This next bit is calculus, and you might not be interested so you 
could jump right to the end...

The volume is calculated by summing up a bunch of really thin disks 
sitting on top of each other:


                     +++++  --------------------+-
                +++         +++                 |
            +++-----------------+++  -----+-    |
         +   |         +         |   +    | dz  |
       +-----+--------/|\--------+-----+ -+-    |
      +                |                +       | R
     +                 |                 +      |
    +                                     +     |
    +                                     +     |
   +                                       +    |
   +            R^2 = x^2 + z^2            +  --+-
   +                                       +
    +                                     +
    +                                     +
     +                                   +
      +                                 +
       +                               +
         +                           +
            +++                 +++
                +++         +++
                     +++++

          z=R
           /
 Volume =  |  Pi*x^2 dz   <===   x^2 = R^2 - z^2
           /
          z=H

          z=R
           /
 Volume =  |  Pi*(R^2 - z^2) dz
           /
          z=H

          z=R              z=R
           /                /
 Volume =  |  Pi*R^2 dz  -  |  Pi*z^2 dz
           /                /
          z=H              z=H

                 z=R           z=R
                  /             /
 Volume = Pi*R^2  |  dz  -  Pi  |  z^2 dz
                  /             /
                 z=H           z=H

                  z=R          z=R
                   |            |
 Volume = Pi*R^2*z | - Pi*z^3/3 |
                   |            |
                  z=H          z=H

 Volume = [ Pi*R^2*R - Pi*R^2*H ] - [ Pi*R^3/3 - Pi*H^3/3 ]

But R=83681/232 and H=R-116=83681/232-116=56769/232

 Volume = [ Pi*(83681/232)^2*83681/232 - Pi*(83681/232)^2*56769/232 ]
        - [ Pi*(83681/232)^3/3 - Pi*(56769/232)^3/3 ]

 Volume = Pi * 12999598/3

 Volume = 13613147.194 cubic feet

That is the volume of the dome part (not including the base).

Now, the total volume is the base volume (Pi * 2528100) plus the 
spherical cap volume (Pi * 12999598/3) and that is:

 TotalVolume = Pi * 20583898/3 cubic feet

 TotalVolume = 21555407.5825 cubic feet

That is a lot of volume!


You'll find an illustration of a spherical cap in the Dr. Math 
Geometric Formulas FAQ at:

   http://mathforum.org/dr.math/faq/formulas/faq.sphere.html      

That page has the formulas, so you don't have to do it the hard way as 
I did...

 Volume = (Pi/6)(3w^2+h^2)h where w=530/2=265 and h=152-36=116
 Volume = (Pi/6)(3*265^2+116^2)116
 Volume = Pi * 12999598/3

Which is the same answer and is useful if you don't care how to solve 
it but just want the answer.

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