Folding angles of 30 and 60 degrees
by Ian Harrison

For those of us who fold paper regularly, creating angles of 90 and 45 degrees is almost second nature. We create a right angle by folding an edge, or a crease, back on itself and we can bisect a right angle (or any other angle) by placing the folds that form the two sides of the angle on top of each other. The mathematics of folding angles of 60 or 30 degrees is only slightly more difficult to understand. (The other angles that paperfolders - especially modular paperfolders - often need to construct are the angles of 108 and 72 degrees, useful in making polyhedra with pentagonal sides and other related forms.)

The method of constructing angles of 60 and 30 degrees by folding is based on the symmetry of an equilateral triangle - one that 3 sides of equal length. To make the explanation clear we will use as an example an equilateral triangle that has sides of 2 units long. We need to look at just one half of it.

The longest side of the right-angle triangle that forms this half, its hypotenuse in the language of geometry, is one side of the original triangle and is therefore still 2 units long. The shortest side of the triangle - at the bottom in the diagram - is half the length of one of the original sides and is therefore only 1 unit long. This ratio of 2:1 between the sides is the key to constructing this kind of triangle - and conequently the angles of 60 and 30 degrees - by folding paper. (For those of you familiar with Euclidean geometry, this is a triangle described by the condition 'right-angle, hypotenuse and side' which is a condition for congruence.)

The diagrams below show the classic way to fold an angle of 60 degrees in the middle of the top edge of a rectangle. To do this, first crease the paper in half to mark the centre-line, then fold the left hand edge to the centre-line to crease the left hand half of the paper into two quarters. (Look at the diagrams - it's easier!) To form the triangle -shown shaded - swing the far top corner across to the quarter way fold, making sure the crease starts at the centre line.

The shaded triangle is an example of the half-equilateral triangle described above. It has a longest side - hypotenuse - of half the length of the top edge of the paper, and a shortest side of one quarter of the length of the top edge of the paper. This creates the ratio of 2:I we are looking for.

The fact that the angle of the shaded triangle at the centre of the top edge is 60 degrees means that the angle next to it along the edge (it's supplement) is 120 degrees, and this is bisected by the fold that we have made. By folding the left hand top corner over to lie along the first folded edge we can create 3 equal angles, each of 60 degrees, at the centre of the top edge.

Folding the other corner across, so that two raw edges lie along each other, bisects the angle of 60 degrees into two angles of 30 degrees each. Of course, we don't have to make the angle of 60 degrees first. We can simply swing the left hand top corner to the quarter way fold (the third diagram below). Again the shaded triangle has the crucial 2:1 ratio between its longest and shortest sides. This time the angle in the triangle at the centre of the top edge is 30 degrees, so the angle next to it, along the edge to the left (its complement) is 60 degrees, and again, this angle is bisected by the fold that we have made.

With small modifications to these ideas we can fold an angle of 60 or 30 degrees at a corner, or at the centre of a square, like this: