The Crafty Part

Things you will need:

• place mat
• sewing machine or needle and thread
• ruler
• scissors
• pre-made handles, canvas strapping, or an old belt.

The placemats I found all measured roughly the same (13 inches by 19 inches) so those are the measurements I will use for this tutorial. The size doesn't matter so much as long as they are rectangular with straight sides and an attractive fabric.


To make the purse you will fold the placemat in half right sides together.


Next you sew up the two sides leaving the top open. (I know that it looks like I didn't sew that drawing all the way to the top, but I did. The hangy threads are just to show that the red is thread).


The next part is trickier. To make the bottom of the purse square you start by pinching one edge together.


From the side it will make an obvious triangle. You simply sew straight across the triangle perpendicular to the side seam at whatever height you prefer in order to get the desired shape.


If you wish at this point you can cut off the excess corner. Turn the purse right side out and the body is complete.

After the body is complete you can add handles following the manufactures directions or use canvas strapping or even an old belt to make one or two handles. You may even want to add a 'frog' or other decorative closure.

Because this project is so quick and so inexpensive you could easily make a different purse to go with any outfit.

The Mathy Part

Next I will show how to use calculus to find the maximum volume of the purse.

To begin with we know that the area of a rectangle (which is how this project starts out) is the length times the width. The volume of our final product (a rectangular prism) will be width(though not the same width) times height times depth.

Because a portion of our original width becomes the side and a portion of our original length becomes the new bottom we need to do some figuring to arrive at a formula that will accurately represent this new shape regardless of the size of triangle we cut out to give it depth.


I used 13 and 19 for the width and length in this tutorial but the actual sizes of you project may vary slightly and the math would need to be re-figured from this point.
So the formula for the volume of the purse made by a 13x19 inch placemat is

In order to find the maximum possible volume we take the Volume formula and find the derivative of it.
If we set the derivative equal to zero and solve for X using the Quadratic Equation we will find the value of X which gives us the maximum volume.
Because there are two possible values of X which make the equation zero we next check each value by plugging it into the original Volume equation.

This makes sense referring back to this picture because the depth is 2X.
If X were 8.14 then the depth would be greater than the available width which is impossible. So the only real answer is X= 2.53 which means that the total depth of our project, in order to obtain the maximum volume is 5.06 inches.

An Aside to Reality

This project didn't turn out exactly as I expected. The purse which results using these measurements, while having the maximum volume, is not the most appealing size. Nor necessarily the most functional.

The one thing not accounted for in these equations is that fabric is not rigid. Therefore the volume of this project varies greatly even if the measurements stay the same.
In fact if the two sides were simply sewn together the project would easily hold things even though mathematically, according to the simple math I have done here, the volume is zero.

There are further calculations which could be done using more advanced calculus (next quarter?) to include this variable. But they are beyond the scope of my knowledge.