
The Procedure: Gravity filtration involves the use of a conical funnel and a circular piece of filter paper. The filtration proceeds most efficiently when the stem of the funnel has a continuous column of liquid with no air bubbles. A funnel is constructed so that the angle of the cone of the funnel (usually 60^{o}, see below) will ideally accomodate filter paper when it is folded exactly in half twice. This means the paper should fit snugly in the funnel when the paper is folded in that way. Real filter paper, however, has a thickness which causes a small deviation from the ideal 60^{o} angle. In order to insure that the paper makes close contact with the funnel somewhere  particularly at the top  we fold the paper so that the angle of the cone that it forms is slightly larger than 60^{o}. The proper folding involves the following: 

1. Fold the filter paper exactly in half. 2. Fold the filter paper a second time, this time with a slight offset. I.e., not exactly in half. 3. Tear a corner from the smaller side of the filter paper. 4. Open the filter paper on the larger side so that the torn corner is on the outside of the cone. 5. Place the filter paper in the funnel pressing the top edge of the cone so that it makes continuous contact with the funnel. The reason for the tear is to maximize the ability of the top edge of the filter paper to make contact with the funnel. (There is effectively only one layers of filter paper where the tear is at the top of the paper,rather than three layers). 

6. Using a wash bottle filled with distilled water, wet the filter paper and carefully press it so that it makes maximum contact with the funnel  particularly around the upper edge in the region of the tear.  
7. Once the filter paper makes maximum contact with the funnel, add an amount of water to cause a continuous column of water to form in the stem of the funnel (This is not always easy to accomplish, but desirable to speed up filtrations.)  
The Theory What is the geometry of the arrangement if we fold the filter paper exactly in half in both the first and second steps? I.e., what is the angle of the cone formed by the filter paper in this case? 

Suppose the filter paper has a radius r. The circumference of the filter paper is 2 π r  
Folding it in half twice exactly results in a quarter circle whose sides are of length r and whose curved segment has a length of 1/4 of the circumference of the original filter paper, or π r / 2.  
Opening this filter paper results in a cone whose circumference is twice the length of this curved segment, or π r.  
Looking at the open cone from the side, the top of the cone has a radius x. But x must be the radius of a circle whose circumference is that of the top of the cone, namely π r. Since the circumference is given by 2 π x, we have  
The sine of the angle a is given by x / r = 1/2. the angle whose sine is 1/2 is 30^{o}. Therefore the half angle of the cone, a, must be 30^{o} and it full angle must be 60^{o}. 

The objective of folding the paper with an offset is to make the cone angle a little larger than 60^{o}. 
Robert F. Schneider (Bob.Schneider at stonybrook.edu)  
Last Update: 20120713 