Bricks to Cover a Steeple

Date: 08/23/97 at 15:43:29
From: Bailey Rector
Subject: Bricks to cover a steeple

Our Literature teacher asked us to figure out how many bricks it would 
take to cover the steeple of a Roman Church in Florence.

The steeple is an octagon with a diameter of 130 feet and a height of 
370 feet.  Each brick is 6 inches by 12 inches.

How do I start to solve this problem?

Date: 08/29/97 at 13:21:05
From: Doctor Rob
Subject: Re: Bricks to cover a steeple

First you need to know the side length of an octagon with diameter 
130 feet. Draw a one-eighth part of an octagon with radii from the 
centerto the vertices. You get an isosceles triangle whose vertex 
angle is 45 degrees and whose base angles are 67.5 degrees each, and 
whose equal sides have length 65 feet.  

Use either the law of sines or the law of cosines to compute the 
length of the base s.  For example, using the law of cosines,

   s^2 = 65^2 + 65^2 - 2*65*65*cos(45 degrees)
       = 4225 + 4225 - 8450*Sqrt[2]/2
       = 65^2*(2 - Sqrt[2])
     s = 65*Sqrt[2 - Sqrt[2]]
       = 49.7488462

Then the area of each face of the octagonal prism is 370*s square feet 
and the lateral area of the octagonal prism will be 8*370*s square 
feet.

Each brick accounts for 1/2 square foot area, so the number of bricks
is 8*370*s/(1/2) = 2*8*370*s, at least. Since this is not an integer,
you will have to round upwards. The number of bricks is at least this
number, if there is no waste at the corners. If you insist that no 
brick surface bend around a corner, you will have to round s up and 
use 50*370*8*2 for the number of bricks, which is somewhat larger.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 08/30/97 at 12:06:13
From: Rick Rector
Subject: Re: Bricks to cover a steeple

Dear Dr Rob,

Thank you for your response.

My dad helped me with the problem and we came up with the same answer 
on the sides of the octagon. We did not have the same answer to the 
rest.

We found the sides of the inside triangle, which is the height of the
center of the steeple of 370 and goes out 65 from the center of the
octogon at the base. This gave us a right triangle with sides 370 and 
65. Using the Pythagorean theorem we squared both numbers and got 
136900 + 4225 = 141125, which has a square root of 375.66607. Dad said 
this should be the length of the cornor of one side of the prism. 

Then to find the area of one side of the prism we found the height. 
We figured the side of the prism was a triangle that had a base of 
49.7488462 and two equal sides of 375.66607. This was cut in half to 
get a right triangle with a base of 24.874423 and hypotenuse of 
375.66607. The square of these are 618.73691 and 141125, so 
141125 - 618.73691 = 140506.26309, which has a square root of 
374.84165 - which gave us a triangle with a base of 49.7488462 and a 
height of 374.84165. 

With area of 1/2bh .5*49.7488462*374.84165 = 9323.9697 square feet 
on each side times 8 sides totalled to 74591.757 square feet.  
The 6" by 12" brick is 1/2 of a square foot, which meant we need two 
bricks per square foot. So we multiplied 74591.757 times 2 to get 
149183.51 bricks or about 149200 bricks.

I turned this in and my teacher said it was not the answer or even
close. He did not give us the answer yet.  

Inside triangle in steeple        ^  Side of steeple
         /|                      / \
        / |                     / | \ 
375.6  /  |                    /  |  \  375.6
      /   | 370               /   |   \
     /    |                  /374.8    \
    /     |                 /     |     \
   /------|                /------|------\
     65                         49.75


What are we doing wrong?

Thank you for your help Dr Rob.

Sincerely,

Bailey Rector

Date: 09/04/97 at 15:08:22
From: Doctor Rob
Subject: Re: Bricks to cover a steeple

First, I think I know what I did wrong.  I assumed from your statement
that the brick part of the steeple was the lateral surface of an 
octagonal prism, that is, the upper and lower faces were congruent
regular octagons with diameter 130 feet, and the sides were all 
rectangles 370 feet high.  

Apparently the steeple is, in fact, an octagonal pyramid, with a 
point, and whose single base is the regular octagon with diameter
130 feet. The area you seek is apparently the area of eight congruent
isosceles triangles whose bases have length equal to the side of the
regular octagon.  

You are correct that the length of the equal sides of the isosceles 
triangles is Sqrt[141125] = 375.666075, and the altitude of each 
triangle is 374.841651 feet. The area is 9323.969857 square feet for
each triangle, or 74,591.758514 square feet altogether.  

Each brick is 1/2 square foot, so 149183.517028 bricks would be 
needed, or 149,184, as you indicated.

It is possible that there is a miscommunication between you (and me) 
and the teacher.  The "height" of 370 feet may mean the height of the 
triangles making up the lateral surface of the pyramid, as opposed to 
the perpendicular height from the vertex to the base. In that case, 
the area would be 8*(1/2)*370*49.7488462 = 73628.292376, and the 
number of bricks twice that, or 147256.584752, rounded up to 147,257, 
or about 147,300. That would account for the different answers found.

I hope this helps, and thanks for straightening me out about the prism 
vs. pyramid mixup.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/