Associated Topics || Dr. Math Home || Search Dr. Math

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.

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/
```
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search