The recent report on the teaching and learning of geometry from the Royal Society and the Joint Mathematical Council has highlighted the central role that geometry should have in school maths both because of its intrinsic importance and interest and because it provides a valuable vehicle for developing students’ powers of proving and problem-solving.

The regular dodecagon shown in figure 1 can be dissected very neatly into 12 equilateral triangles and 12 rhombuses all with the same edge lengths. This amazing dodecagon offers great opportunities at all levels, from primary school to the sixth-form, to apply geometrical thinking to topics such as angle properties of polygons, ruler and compass constructions, symmetry, fractions, areas and trigonometry.

We can easily see that the acute angles of the rhombuses are 30 LESS THAN , because there are 12 such equal angles at the centre, so each is 112 of 360 LESS THAN . Now look at the three angles at one of the inner ring of points which link the rhombuses. Since the angle of the equilateral triangle is 60 LESS THAN , the two obtuse angles contributed by the rhombus must add up to 300 LESS THAN , so each must be 150 LESS THAN . In a similar way, three angles meet at each vertex of the dodecagon, so we can calculate the interior angle as 60 LESS THAN + 30 LESS THAN + 60 LESS THAN , which is 150 LESS THAN again.

Once we know the angles of the rhombus it is easy to construct the diagram using ruler and compasses - simply draw a circle with the same radius as the edges you want for the decagon, mark six equally spaced points round it, join them to the centre and then bisect one of the 60 LESS THAN angles formed at the centre. Now use the bisector to find a point on the circle midway between two of the first six points and use that to mark the other midpoints. Finally, the outer vertex of each rhombus is found by drawing arcs from each pair of points on this inner circle of points. As a further challenge in using the angle properties, try creating the picture on a computer screen using LOGO or dynamic geometry software.

One of the intriguing properties of the dissection is that the 12 triangles and 12 rhombuses can be rearranged to make the dodecagon in a host of different ways. Figure 2 shows one of the many possibilities. You will notice that the 150 LESS THAN angle is very conveniently used at some of the vertices of the dodecagon while the others are made up of three 30 LESS THAN angles and one of 60 LESS THAN . Looking at the symmetry is interesting: figure 1 has 12 lines of symmetry and rotational symmetry of order 12 and figure 2 displays rotational symmetry of order 6 and has no line symmetry. Colouring the rhombuses and triangles in different ways adds a further element to the search for symmetry. Looking for some of the alternative configurations and examining their angle and symmetry properties is a rewarding task. The initial configuration can be drawn on thin card and cut out to give the 24 shapes. Using card of several different colours makes the configurations even more attractive and using shapes drawn on coloured gummed paper is a simple way to create attractive diagrams for display.

Yet further possibilities are possible if some squares with the same edge-length are introduced. Figure 3 shows one example: angle properties and symmetry can be looked at again, but it also gives a simple way to find the area of the rhombus. There are 12 equilateral triangles as before, but only four rhombuses. Since the other eight rhombuses have been replaced by four squares, it follows that the area of one of these rhombuses is half the area of a square.

This way to see the area is much simpler than a trigonometrical method using sin 30 LESS THAN , although that is of importance because it generalises to all rhombuses. To find the area of the equilateral triangle, we need Pythagoras’s theorem to show that the height is 12C3 when the edges are of unit length. The area is then 14C3, or approximately 0.43, slightly less than the area of 12 for the rhombus. The ratio of the areas of the two shapes is C3:2 and the area of the dodecagon is 6+3C3.

The biggest surprise is provided by figure 4. Placing four equilateral triangles and eight half rhombuses round the edges of the dodecagon gives a square. A careful check on all the vertices shows that the pieces do fit together to give straight edges and right angles at the corners. The square consists of 16 equilateral triangles and 16 rhombuses, whereas there are 12 of each in the dodecagon. So, the area of the dodecagon is three quarters of the area of the square and that has an interesting consequence. If r is the radius of the circle circumscribing the dodecagon, the square has edges of 2r and an area of 4r2. It follows that the area of the dodecagon is 3r2, a little less than the familiar 9r2 for the area of the circle!

The square tile is known as Kuersch k’s Tile, after the Hungarian mathematician J“zsef Kuersch k (1864-1933). I originally came across it more than 20 years ago in an article by J L Anderson and K Sydel in the Mathematical Gazette, which has been reproduced in the Mathematical Association’s book of readings which I edited (with Charlie Stripp) called Pig and Other Tales (www.m-a.org.uk).

This simple dissection should be more widely known because its many surprising properties offer an interesting variety of tasks and challenges for a wide range of students, besides providing some attractive material for displaying on classroom walls.

* To receive a copy of the report Teaching and Learning Geometry 11-19 from the Royal SocietyJoint Mathematical Council send an A4 self-addressed envelope to: The Royal Society, 6 Carlton House Terrace, London SW1Y 5AG; tel: 020 7451 2572; e-mail: education@royalsoc.ac.uk; web: www.royalsoc.ac.uk Doug French is a lecturer in education at the University of Hull and is chair of the Mathematical Association’s Teaching Committee. E-mail: d.w.french@hull.ac.uk