Painting a Column

Date: 07/13/97 at 15:39:34
From: Cathy Ryan
Subject: Painting a Column

If I have a column 9.5 feet long and 8 inches in diameter, how do I 
figure how much paint I need to cover the outside area? 

Date: 07/16/97 at 04:40:39
From: Doctor Sydney
Subject: Re: Painting a Column

To help you understand how to do this problem, let's make a model:

Get a piece of paper and roll it up to look like a column so that the 
long side of the paper is the height of the column.  The circumference 
of the column is the distance around the column and corresponds to the 
width of the paper. 

You can see how to solve this problem more easily when you now unroll 
the paper. The height of the column corresponds to the length of the 
long side of the paper. The circumference corresponds to the width of 
the paper.  

So, given this information, how can we calculate the surface area? 
First, let's assume that we don't care about painting the top and 
bottom of the column (we'll deal with this case in a minute). Then 
all we have to worry about is the surface area on the portion of the 
column that we "rolled out" into a rectangle.  

The previous demonstration showed that the surface area of a column 
(without its base and top) is equal to the area of the rectangle with 
height equal to the height of the column and width equal to the 
circumference of the column. So, if we call the height of the column h 
and the circumference C, then the surface area of the column (and the 
area of the corresponding rectangle) will be A = hC.  Can you figure 
out the surface area of this part of the column from here? You just 
need to know how to find circumference given diamenter, and you have 
to be able to convert inches to feet or vice versa. If you need more 
help, please write back and I can help out. 

Now, if you want to paint the top or bottom of the column, we need to 
consider these separately. Suppose you wanted to paint the top. Well, 
look at the top of the column. It is really just a circle with 
diameter 8 in., right? The equation for the area of a circle is 
A = pi*r^2 where r is the radius of the circle. I'll leave it to you 
to calculate what the area of your base and top are. Again, if you 
have trouble, please do let me know.

Once you have calculated the surface area for the middle part of the 
column, the base, and the top, add these together to figure out what 
is the total surface area of the column.  

Next you need to know how much paint you need to cover this much area.  
Call a paint store and ask them how much surface area a gallon will 
cover. Then, make sure that the store's surface area and your surface 
area are in the same units - if they aren't, convert them so that they 
are, and then divide the surface area by this number and you will know 
how much paint you need.

Hope that helped! Please write back if you would like more guidance.  

-Doctors Bernard and Sydney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/