“Dividing a Sheet of Paper into Three Equal Thirds with one Equilateral Triangle”

by

Christopher R. Whitworth

You have been given a blank sheet of paper (8.50 cm x 11.01 cm), and have been instructed to divide the sheet of paper into thirds (in area) while one of the thirds represents an equilateral triangle.  The blank sheet of paper is shown below.  (Note:  Some values may not be completely accurate and may differ by no more than 1/50th of a centimeter).

The next step is to create a midpoint between each of the two line segments, FH and GI.  Then connect the two midpoints with a dotted line that divides the sheet of paper in half.

Then, you should create two lines that are 8.50 cm in length each that connects point G to the red-dotted line and I to the red-dotted line.  The reason you are selecting 8.50 cm as your value is because you want at least one equilateral triangle (three equal sides and angles).  The bottom of the sheet of paper is already 8.50 cm, so if you make the lines 8.50 cm and have them intersect at one point, then you have created an equilateral triangle.

Next, you will want to hide the section of the dotted line that lies just below point J.

The next step will be to fill in each of the three sections a different color so that you can distinguish between the three sections.

Then, you will calculate the area of each section by using either Geometers Sketchpad 4.0 or a ruler and the formula for area (A = L x W).

Notice that the area of each section is equal to 31.19 cm.  This means that the area of each triangle is equal.  I have shown you how to divide a sheet of paper into three sections with equal areas including one equilateral triangle.  Below is the pattern on how to fold the paper to determine the three equal sections.

You have just folded the paper in half.  Now, unfold the paper and fold the right corner to the fold in the center.  Then, draw a point on the paper where the corner of the paper touched the fold.  Do not fold the corner back to the original position.  Next, you will want to repeat this process with the left corner and the point on the fold in the center.

Finally, you will notice a shape that looks similar to the one below.  Once you have constructed this figure, unfold the paper to its original position and measure the area of each section.  You will notice that the area in each figure will be equal to one other.

Once you have determined that each figure’s area is equal to one another, then you can prove that you can divide a sheet of paper that is 8.50 cm x 11.00 cm into three sections with equal areas while including an equilateral triangle as one of the sections.

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