Octagon Star

To build the octagon star you need eight pieces of origami paper. You may use either 4 pieces each of two colors or 2 pieces each of four colors. Each piece of origami paper will be folded in the same manner. You will then assemble the folded pieces together, according to the following instructions.

Folding each piece of origami paper:

Fold the paper in half and then unfold it. Fold it so that the colored sides remain on the outside.

Rotate the paper 90 degrees, fold in half and then unfold. Again, fold so that the color remains on the outside.

Fold the top two corners down to the center.

Fold point A down to point C to make the crease in the middle of the left square. Do not crease past the center of the piece of origami paper. Unfold and then do the same on the other square by folding point B down to point C. Unfold this fold. Your origami paper should then look like the following picture.

Rotate the paper 90 degrees and fold it in half.

Push the fold inside by pushing point C to point A. This will require reversing the fold that goes down the center of the white rectangle in the previous picture. You will then have a parallelogram. One end of the parallelogram will be made up of two flaps forming a valley.

Assembling the pieces.

Place one parallelogram inside another of a different color as the following picture indicates. The short side of the second parallelogram should go inside the valley of the first parallelogram.

Fold the tails of the first parallelogram over the second to lock them together. The following picture indicates one of the tails folded into the valley of the second parallelogram.

Repeat this procedure until you have locked all 8 parallelograms to form the resulting octagonal ring. When you hook the 8th with the first, make sure that you fold the tails over only the first parallelogram and not the first and second (you'll have to look carefully at the resulting shape to follow this instruction). The resulting figure should slide together fairly easily to form the star at the top of the assignment.

Question: Determine, as a fraction of the length of the side of the origami paper, the lengths of the sides of the resulting parallelogram along with the length of its diagonal. What is the area of the parallelogram, as a fraction of the area of the origami paper?