This is how I did it.

1. Mark points A, B, C, D, E equally spaced around the circle. Mark point F diametrically opposite point A.

2. Fold the circle along line AF so that point B lies on top of point E, and point C lies on top of point D.

3. Fold the circle again so that point C/D lies on top of point B/E. This will put point F halfway between A and B/C/D/E.

4. Make the first cut from point B/C/D/E to the center of the circle. This will divide it into 4 parts, 3 of which are equal fifths.

5. Make the second cut from point A to the center of the circle. This will cut the piece of size 2/5 into two pieces of size 1/5.

1. Mark points A, B, C, D, E equally spaced around the circle. Mark point F diametrically opposite point A.

2. Fold the circle along line AF so that point B lies on top of point E, and point C lies on top of point D.

3. Fold the circle again so that point C/D lies on top of point B/E. This will put point F halfway between A and B/C/D/E.

4. Make the first cut from point B/C/D/E to the center of the circle. This will divide it into 4 parts, 3 of which are equal fifths.

5. Make the second cut from point A to the center of the circle. This will cut the piece of size 2/5 into two pieces of size 1/5.